id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-13000 | Order of Product of Entire Function with Polynomial | Let $f: \C \to \C$ be an entire function of order $\omega$.
Let $P: \C \to \C$ be a nonzero polynomial.
Then $f \cdot P$ has order $\omega$. | If $f=0$, then $f \cdot P = 0$.
Thus the claim is trivial.
Therefore suppose that $f$ is not identically zero.
By Order of Product of Entire Functions and Polynomial has Order Zero, $f\cdot P$ has order at most $\omega$.
By Limit at Infinity of Polynomial, there exist $r, \delta > 0$ such that $\size {\map P z} \ge \de... | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\omega$.
Let $P: \C \to \C$ be a [[Definition:Zero Polynomial|nonzero]] [[Definition:Complex Polynomial Function|polynomial]].
Then $f \cdot P$ has [[Definition:Order of Entire Function|order]] $... | If $f=0$, then $f \cdot P = 0$.
Thus the claim is trivial.
Therefore suppose that $f$ is not identically zero.
By [[Order of Product of Entire Functions]] and [[Polynomial has Order Zero]], $f\cdot P$ has order at most $\omega$.
By [[Limit at Infinity of Polynomial]], there exist $r, \delta > 0$ such that $\size {... | Order of Product of Entire Function with Polynomial | https://proofwiki.org/wiki/Order_of_Product_of_Entire_Function_with_Polynomial | https://proofwiki.org/wiki/Order_of_Product_of_Entire_Function_with_Polynomial | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Null Polynomial",
"Definition:Polynomial Function/Complex",
"Definition:Order of Entire Function"
] | [
"Order of Product of Entire Functions",
"Polynomial has Order Zero",
"Limit at Infinity of Polynomial",
"Maximum Modulus Principle",
"Definition:Contradiction",
"Definition:Order of Entire Function"
] |
proofwiki-13001 | Order of Shifted Entire Function | Let $f: \C \to \C$ be an entire function of order $\alpha$.
Let $a \in \C$.
Then $\map f {z + a}$ has order $\alpha$. | We shall verify {{Defof|Order of Entire Function|index=3}}.
Let $\map g z := \map f {z + a}$ for $z \in \C$.
Then for all $R > \cmod a$:
:$\ds \max_{\cmod z \mathop \le R \mathop - \cmod a} \cmod {\map f z} \le \max_{\cmod z \mathop \le R} \cmod {\map g z} \le \max_{\cmod z \mathop \le R \mathop + \cmod a} \cmod {\map ... | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\alpha$.
Let $a \in \C$.
Then $\map f {z + a}$ has [[Definition:Order of Entire Function|order]] $\alpha$. | We shall verify {{Defof|Order of Entire Function|index=3}}.
Let $\map g z := \map f {z + a}$ for $z \in \C$.
Then for all $R > \cmod a$:
:$\ds \max_{\cmod z \mathop \le R \mathop - \cmod a} \cmod {\map f z} \le \max_{\cmod z \mathop \le R} \cmod {\map g z} \le \max_{\cmod z \mathop \le R \mathop + \cmod a} \cmod {\ma... | Order of Shifted Entire Function | https://proofwiki.org/wiki/Order_of_Shifted_Entire_Function | https://proofwiki.org/wiki/Order_of_Shifted_Entire_Function | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Order of Entire Function"
] | [
"Properties of Limit at Infinity of Real Function/Product Rule",
"Squeeze Theorem"
] |
proofwiki-13002 | Zerofree Entire Function of Finite Order is Exponential of Polynomial | Let $f: \C \to \C$ be an entire function of finite order.
Let $f$ have no zeroes.
Then $f = \exp P$ for some polynomial $P$. | This is an immediate consequence of Hadamard Factorization Theorem.
{{qed}}
Category:Entire Functions
5u1070qxsmw80zn68uyesvu3g35yv5s | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|finite order]].
Let $f$ have no [[Definition:Zero of Function|zeroes]].
Then $f = \exp P$ for some [[Definition:Polynomial|polynomial]] $P$. | This is an immediate consequence of [[Hadamard Factorization Theorem]].
{{qed}}
[[Category:Entire Functions]]
5u1070qxsmw80zn68uyesvu3g35yv5s | Zerofree Entire Function of Finite Order is Exponential of Polynomial | https://proofwiki.org/wiki/Zerofree_Entire_Function_of_Finite_Order_is_Exponential_of_Polynomial | https://proofwiki.org/wiki/Zerofree_Entire_Function_of_Finite_Order_is_Exponential_of_Polynomial | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Root of Mapping",
"Definition:Polynomial"
] | [
"Hadamard Factorization Theorem",
"Category:Entire Functions"
] |
proofwiki-13003 | Borel-Carathéodory Lemma | Let $D \subset \C$ be an open set with $0 \in D$.
Let $R > 0$ be such that the open disk $\map B {0, R} \subset D$.
Let $f: D \to \C$ be analytic with $\map f 0 = 0$.
Let $\map \Re {\map f z} \le M$ for $\cmod z \le R$.
Let $0 < r < R$.
Then for $\cmod z \le r$:
{{begin-eqn}}
{{eqn | n = 1
| l = \cmod {\map f z}
... | {{begin-eqn}}
{{eqn | l = \map f z
| r = \sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n
| c = Taylor Series of Holomorphic Function
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n
| c = as $\map f 0 = 0$
}}
{{end-eqn}} | Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open set]] with $0 \in D$.
Let $R > 0$ be such that the [[Definition:Open Complex Disk|open disk]] $\map B {0, R} \subset D$.
Let $f: D \to \C$ be [[Definition:Analytic Complex Function|analytic]] with $\map f 0 = 0$.
Let $\map \Re {\map f z} \le M$ f... | {{begin-eqn}}
{{eqn | l = \map f z
| r = \sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n
| c = [[Taylor Series of Holomorphic Function]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\map {f^{\paren n} } 0} {n!} z^n
| c = as $\map f 0 = 0$
}}
{{end-eqn}} | Borel-Carathéodory Lemma | https://proofwiki.org/wiki/Borel-Carathéodory_Lemma | https://proofwiki.org/wiki/Borel-Carathéodory_Lemma | [
"Complex Analysis"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Complex Disk/Open",
"Definition:Analytic Function/Complex Plane"
] | [
"Taylor Series of Holomorphic Function"
] |
proofwiki-13004 | Order of Sum of Entire Functions | Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$.
Then $f + g$ has order at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$. | {{MissingLinks|Each step of this proof needs to be backed up with evidence: a link to a result or a definition, preferably.}}
Let $\map f z = \map \OO {e^{\cmod z^m} }$ and $\map g z = \map \OO {e^{\cmod z^k} }$.
{{WLOG}}, let $m \ge k$.
Then it follows that $\map f z + \map g z = \map \OO {e^{\cmod z^m} }$.
Then the o... | Let $f, g: \C \to \C$ be [[Definition:Entire Function|entire functions]] of [[Definition:Order of Entire Function|order]] $\alpha$ and $\beta$.
Then $f + g$ has [[Definition:Order of Entire Function|order]] at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$. | {{MissingLinks|Each step of this proof needs to be backed up with evidence: a link to a result or a definition, preferably.}}
Let $\map f z = \map \OO {e^{\cmod z^m} }$ and $\map g z = \map \OO {e^{\cmod z^k} }$.
{{WLOG}}, let $m \ge k$.
Then it follows that $\map f z + \map g z = \map \OO {e^{\cmod z^m} }$.
Then t... | Order of Sum of Entire Functions | https://proofwiki.org/wiki/Order_of_Sum_of_Entire_Functions | https://proofwiki.org/wiki/Order_of_Sum_of_Entire_Functions | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Order of Entire Function"
] | [] |
proofwiki-13005 | Exponent of Convergence is Less Than Order | Let $f: \C \to \C$ be an entire function.
Let $\omega$ be its order.
Let $\tau$ be its exponent of convergence.
Then $\tau \le \omega$. | We may assume $\map f 0 \ne 0$.
Let $f$ have finitely many zeroes.
Then:
:$\tau = 0 \le \omega$
{{qed|lemma}}
Let $f$ have infinitely many zeroes.
Let $\sequence {a_n}$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity and ordered by increasing modulus.
Let $r_n = \size {a_n}$ and $R_n = 2 \s... | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $\omega$ be its [[Definition:Order of Entire Function|order]].
Let $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
Then $\tau \le \omega$. | We may assume $\map f 0 \ne 0$.
Let $f$ have [[Definition:Finite Set|finitely many]] zeroes.
Then:
:$\tau = 0 \le \omega$
{{qed|lemma}}
Let $f$ have [[Definition:Infinite Set|infinitely many]] zeroes.
Let $\sequence {a_n}$ be the [[Definition:Sequence|sequence]] of nonzero zeroes of $f$, repeated according to [[D... | Exponent of Convergence is Less Than Order | https://proofwiki.org/wiki/Exponent_of_Convergence_is_Less_Than_Order | https://proofwiki.org/wiki/Exponent_of_Convergence_is_Less_Than_Order | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Exponent of Convergence"
] | [
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Sequence",
"Definition:Multiplicity (Complex Analysis)",
"Jensen's Inequality (Complex Analysis)",
"Definition:Order of Entire Function",
"Convergence of P-Series"
] |
proofwiki-13006 | Class is Proper iff Bijection from Class to Proper Class/Corollary | $A$ is proper {{iff}} there exists a bijection from $P$ to $A$. | {{NotZFC}}
From Biconditional is Transitive and Class is Proper iff Bijection from Class to Proper Class, it suffices to show that:
:There exists a bijection from $A$ to $P$ {{iff}} there exists a bijection from $P$ to $A$.
The rest follows from Inverse of Bijection is Bijection.
{{qed}}
Category:Gödel-Bernays Class Th... | $A$ is [[Definition:Proper Class|proper]] {{iff}} there exists a [[Definition:Class Bijection|bijection]] from $P$ to $A$. | {{NotZFC}}
From [[Biconditional is Transitive]] and [[Class is Proper iff Bijection from Class to Proper Class]], it suffices to show that:
:There exists a [[Definition:Class Bijection|bijection]] from $A$ to $P$ {{iff}} there exists a [[Definition:Class Bijection|bijection]] from $P$ to $A$.
The rest follows from ... | Class is Proper iff Bijection from Class to Proper Class/Corollary | https://proofwiki.org/wiki/Class_is_Proper_iff_Bijection_from_Class_to_Proper_Class/Corollary | https://proofwiki.org/wiki/Class_is_Proper_iff_Bijection_from_Class_to_Proper_Class/Corollary | [
"Gödel-Bernays Class Theory",
"Class Mappings"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Bijection/Class Theory"
] | [
"Biconditional is Transitive",
"Class is Proper iff Bijection from Class to Proper Class",
"Definition:Bijection/Class Theory",
"Definition:Bijection/Class Theory",
"Inverse of Bijection is Bijection",
"Category:Gödel-Bernays Class Theory",
"Category:Class Mappings"
] |
proofwiki-13007 | Length of God's Algorithm for Sam Loyd's Fifteen Puzzle | The length of God's Algorithm for Sam Loyd's Fifteen Puzzle is $52$. | {{ProofWanted|A lot of background material needed first.}} | The [[Definition:Length of Algorithm|length]] of [[God's Algorithm for Sam Loyd's Fifteen Puzzle]] is $52$. | {{ProofWanted|A lot of background material needed first.}} | Length of God's Algorithm for Sam Loyd's Fifteen Puzzle | https://proofwiki.org/wiki/Length_of_God's_Algorithm_for_Sam_Loyd's_Fifteen_Puzzle | https://proofwiki.org/wiki/Length_of_God's_Algorithm_for_Sam_Loyd's_Fifteen_Puzzle | [
"Recreational Mathematics",
"52"
] | [
"Definition:Length of Algorithm",
"God's Algorithm for Sam Loyd's Fifteen Puzzle"
] | [] |
proofwiki-13008 | Injection from Proper Class to Class | Let $A$ be a class.
Let $P$ be a proper class.
Let $f: P \to A$ be an injection.
Then $A$ is proper. | {{NotZFC}}
{{AimForCont}} $A$ is not proper.
Then it must be a set.
By Injection to Image is Bijection, it follows that the restriction $f \restriction_{P \times f \sqbrk P}$ is a bijection.
By the corollary of Class is Proper iff Bijection from Class to Proper Class, $f \sqbrk P$ is proper.
But since $f \sqbrk P \subs... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $P$ be a [[Definition:Proper Class|proper class]].
Let $f: P \to A$ be an [[Definition:Class Injection|injection]].
Then $A$ is [[Definition:Proper Class|proper]]. | {{NotZFC}}
{{AimForCont}} $A$ is not [[Definition:Proper Class|proper]].
Then it must be a [[Definition:Set|set]].
By [[Injection to Image is Bijection]], it follows that the [[Definition:Restriction of Mapping|restriction]] $f \restriction_{P \times f \sqbrk P}$ is a [[Definition:Class Bijection|bijection]].
By ... | Injection from Proper Class to Class | https://proofwiki.org/wiki/Injection_from_Proper_Class_to_Class | https://proofwiki.org/wiki/Injection_from_Proper_Class_to_Class | [
"Gödel-Bernays Class Theory",
"Class Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Injection/Class Theory",
"Definition:Class (Class Theory)/Proper Class"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Set",
"Injection to Image is Bijection",
"Definition:Restriction/Mapping",
"Definition:Bijection/Class Theory",
"Class is Proper iff Bijection from Class to Proper Class/Corollary",
"Definition:Class (Class Theory)/Proper Class",
"Definition:... |
proofwiki-13009 | Polynomial has Order Zero | Let $P: \C \to \C$ be a polynomial function.
Then $P$ has order $0$. | Follows from a slightly stronger result than Limit at Infinity of Polynomial over Complex Exponential.
{{ProofWanted}} | Let $P: \C \to \C$ be a [[Definition:Complex Polynomial Function|polynomial function]].
Then $P$ has [[Definition:Order of Entire Function|order]] $0$. | Follows from a slightly stronger result than [[Limit at Infinity of Polynomial over Complex Exponential]].
{{ProofWanted}} | Polynomial has Order Zero | https://proofwiki.org/wiki/Polynomial_has_Order_Zero | https://proofwiki.org/wiki/Polynomial_has_Order_Zero | [
"Entire Functions"
] | [
"Definition:Polynomial Function/Complex",
"Definition:Order of Entire Function"
] | [
"Limit at Infinity of Polynomial over Complex Exponential"
] |
proofwiki-13010 | Period of Reciprocal of 53 is One Quarter of Maximal | The decimal expansion of the reciprocal of $53$ has $\dfrac 1 4$ the maximum period, that is: $13$:
:$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ | From Reciprocal of $53$:
{{:Reciprocal of 53}}
Counting the digits, it is seen that this has a period of recurrence of $13$.
From Maximum Period of Reciprocal of Prime, the maximum period of recurrence of $\dfrac 1 p$ is $p - 1$.
We have that:
:$13 = \dfrac {53 - 1} 4$
Hence the result.
{{qed}} | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $53$ has $\dfrac 1 4$ the maximum [[Definition:Period of Recurrence|period]], that is: $13$:
:$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ | From [[Reciprocal of 53|Reciprocal of $53$]]:
{{:Reciprocal of 53}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $13$.
From [[Maximum Period of Reciprocal of Prime]], the maximum [[Definition:Period of Recurrence|period of recurrence]] of $\dfrac 1 p$ is ... | Period of Reciprocal of 53 is One Quarter of Maximal | https://proofwiki.org/wiki/Period_of_Reciprocal_of_53_is_One_Quarter_of_Maximal | https://proofwiki.org/wiki/Period_of_Reciprocal_of_53_is_One_Quarter_of_Maximal | [
"53",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 53",
"Definition:Basis Expansion/Recurrence/Period",
"Maximum Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-13011 | Order is Maximum of Exponent of Convergence and Degree | Let $f: \C \to \C$ be an entire function.
Let $\omega$ be its order.
Let $\tau$ be its exponent of convergence.
Let $h$ be the degree of the polynomial in its canonical factorization.
Then:
:$\omega = \map \max {\tau, h}$ | By Exponent of Convergence is Less Than Order:
:$\tau \le \omega$
By Hadamard Factorization Theorem:
:$h \le \omega$
Thus:
:$\map \max {\tau, h} \le \omega$
By Order is Less Than Maximum of Exponent of Convergence and Degree:
:$\omega \le \map \max {\tau, h}$
{{qed}} | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $\omega$ be its [[Definition:Order of Entire Function|order]].
Let $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
Let $h$ be the [[Definition:Degree of Polynomial|degree]] of the [[Definition:Polynomial|polyno... | By [[Exponent of Convergence is Less Than Order]]:
:$\tau \le \omega$
By [[Hadamard Factorization Theorem]]:
:$h \le \omega$
Thus:
:$\map \max {\tau, h} \le \omega$
By [[Order is Less Than Maximum of Exponent of Convergence and Degree]]:
:$\omega \le \map \max {\tau, h}$
{{qed}} | Order is Maximum of Exponent of Convergence and Degree | https://proofwiki.org/wiki/Order_is_Maximum_of_Exponent_of_Convergence_and_Degree | https://proofwiki.org/wiki/Order_is_Maximum_of_Exponent_of_Convergence_and_Degree | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Exponent of Convergence",
"Definition:Degree of Polynomial",
"Definition:Polynomial",
"Definition:Hadamard's Canonical Factorization"
] | [
"Exponent of Convergence is Less Than Order",
"Hadamard Factorization Theorem",
"Order is Less Than Maximum of Exponent of Convergence and Degree"
] |
proofwiki-13012 | Recurring Parts of Multiples of Reciprocal of 53 | The multiples of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ sets of equal size:
:one where the digits of the recurring part consists of a cyclic permutation of $01886 \, 79245 \, 283$
:one where the digits of the recurring part consists of a cyclic permutation of $03773 \, 58490 ... | {{begin-eqn}}
{{eqn | l = 1 / 53
| r = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3
| c =
}}
{{eqn | l = 2 / 53
| r = 0 \cdotp \dot 03773 \, 58490 \, 56 \dot 6
| c =
}}
{{eqn | l = 3 / 53
| r = 0 \cdotp \dot 05660 \, 37735 \, 84 \dot 9
| c =
}}
{{eqn | l = 4 / 53
| r = 0 \cdotp... | The [[Definition:Rational Multiplication|multiples]] of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ [[Definition:Set|sets]] of equal [[Definition:Cardinality|size]]:
:one where the [[Definition:Digit|digits]] of the [[Definition:Recurring Part|recurring part]] consists of a [[Def... | {{begin-eqn}}
{{eqn | l = 1 / 53
| r = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3
| c =
}}
{{eqn | l = 2 / 53
| r = 0 \cdotp \dot 03773 \, 58490 \, 56 \dot 6
| c =
}}
{{eqn | l = 3 / 53
| r = 0 \cdotp \dot 05660 \, 37735 \, 84 \dot 9
| c =
}}
{{eqn | l = 4 / 53
| r = 0 \cdotp... | Recurring Parts of Multiples of Reciprocal of 53 | https://proofwiki.org/wiki/Recurring_Parts_of_Multiples_of_Reciprocal_of_53 | https://proofwiki.org/wiki/Recurring_Parts_of_Multiples_of_Reciprocal_of_53 | [
"Number Theory",
"53"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Cardinality",
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Cyclic Permut... | [] |
proofwiki-13013 | Order of Reciprocal of Entire Function | Let $f: \C \to \C$ be an entire function of order $\rho$.
Let $f$ have no zeroes.
Then $1/f$ has order $\rho$. | By Zerofree Analytic Function on Simply Connected Set has Logarithm, there exists an entire function $g$ with $f = \exp g$.
{{ProofWanted}} | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]] of [[Definition:Order of Entire Function|order]] $\rho$.
Let $f$ have no zeroes.
Then $1/f$ has [[Definition:Order of Entire Function|order]] $\rho$. | By [[Zerofree Analytic Function on Simply Connected Set has Logarithm]], there exists an [[Definition:Entire Function|entire function]] $g$ with $f = \exp g$.
{{ProofWanted}} | Order of Reciprocal of Entire Function | https://proofwiki.org/wiki/Order_of_Reciprocal_of_Entire_Function | https://proofwiki.org/wiki/Order_of_Reciprocal_of_Entire_Function | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Order of Entire Function"
] | [
"Zerofree Analytic Function on Simply Connected Set has Logarithm",
"Definition:Entire Function"
] |
proofwiki-13014 | Jacobi's Necessary Condition/Dependent on N Functions | Let $J$ be a functional, such that:
:$J \sqbrk {\mathbf y} = \ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector.
Let $\map {\mathbf y} x$ correspond to the minimum of $J$.
Let the $N\times N$ matrix $\mathbf P = F_{y_i' y_j... | By Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum, $J$ is minimised by $y = \map {\mathbf {\hat y} } x$ if:
:$\delta^2 J \sqbrk {\mathbf {\hat y}; \mathbf h} \ge 0$
for all admissable real functions $\mathbf h$.
By lemma 1 of Legendre's Condition:
:$\ds \delta^2 J \sqbrk {... | Let $J$ be a [[Definition:Real Functional|functional]], such that:
:$J \sqbrk {\mathbf y} = \ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector.
Let $\map {\mathbf y} x$ correspond to the [[Definition:Minimum Value of Fun... | By [[Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum]], $J$ is minimised by $y = \map {\mathbf {\hat y} } x$ if:
:$\delta^2 J \sqbrk {\mathbf {\hat y}; \mathbf h} \ge 0$
for all admissable [[Definition:Real Function|real functions]] $\mathbf h$.
By [[Legendre's Condition... | Jacobi's Necessary Condition/Dependent on N Functions | https://proofwiki.org/wiki/Jacobi's_Necessary_Condition/Dependent_on_N_Functions | https://proofwiki.org/wiki/Jacobi's_Necessary_Condition/Dependent_on_N_Functions | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Minimum Value of Functional",
"Definition:Real Interval/Open",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)"
] | [
"Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum",
"Definition:Real Function",
"Legendre's Condition/Lemma 1/Dependent on N Functions",
"Nonnegative Quadratic N function dependent Functional implies no Interior Conjugate Points",
"Definition:Conjugate Point (Calc... |
proofwiki-13015 | Probability of no 2 People out of 53 Sharing the Same Birthday | Let there be $53$ people in a room.
The probability that no $2$ of them have the same birthday is approximately $\dfrac 1 {53}$. | {{Refactor|The analyis has already been performed in Birthday Paradox. Extract that general case and make it a theorem, and then introduce Birthday Paradox and this page as examples.|level = medium}}
Let there be $n$ people in the room.
Let $\map p n$ be the probability that no two people in the room have the same birt... | Let there be $53$ people in a room.
The [[Definition:Probability|probability]] that no $2$ of them have the same birthday is approximately $\dfrac 1 {53}$. | {{Refactor|The analyis has already been performed in [[Birthday Paradox]]. Extract that general case and make it a theorem, and then introduce [[Birthday Paradox]] and this page as examples.|level = medium}}
Let there be $n$ people in the room.
Let $\map p n$ be the [[Definition:Probability|probability]] that no two ... | Probability of no 2 People out of 53 Sharing the Same Birthday | https://proofwiki.org/wiki/Probability_of_no_2_People_out_of_53_Sharing_the_Same_Birthday | https://proofwiki.org/wiki/Probability_of_no_2_People_out_of_53_Sharing_the_Same_Birthday | [
"Probability Theory",
"53"
] | [
"Definition:Probability"
] | [
"Birthday Paradox",
"Birthday Paradox",
"Definition:Probability",
"Definition:Leap Year",
"Definition:Time/Unit/Year",
"Definition:Probability",
"Definition:Probability",
"Definition:Probability",
"Definition:Probability",
"Definition:Probability"
] |
proofwiki-13016 | Triangular Fibonacci Numbers | The only Fibonacci numbers which are also triangular are:
:$0, 1, 3, 21, 55$
{{OEIS|A039595}} | {{begin-eqn}}
{{eqn | l = 0
| r = \dfrac {0 \times 1} 2
}}
{{eqn | l = 1
| r = \dfrac {1 \times 2} 2
}}
{{eqn | l = 3
| r = \dfrac {2 \times 3} 2
| rr= = 1 + 2
}}
{{eqn | l = 21
| r = \dfrac {6 \times 7} 2
| rr= = 8 + 13
}}
{{eqn | l = 55
| r = \dfrac {10 \times 11} 2
| r... | The only [[Definition:Fibonacci Number|Fibonacci numbers]] which are also [[Definition:Triangular Number|triangular]] are:
:$0, 1, 3, 21, 55$
{{OEIS|A039595}} | {{begin-eqn}}
{{eqn | l = 0
| r = \dfrac {0 \times 1} 2
}}
{{eqn | l = 1
| r = \dfrac {1 \times 2} 2
}}
{{eqn | l = 3
| r = \dfrac {2 \times 3} 2
| rr= = 1 + 2
}}
{{eqn | l = 21
| r = \dfrac {6 \times 7} 2
| rr= = 8 + 13
}}
{{eqn | l = 55
| r = \dfrac {10 \times 11} 2
| r... | Triangular Fibonacci Numbers | https://proofwiki.org/wiki/Triangular_Fibonacci_Numbers | https://proofwiki.org/wiki/Triangular_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Triangular Numbers"
] | [
"Definition:Fibonacci Number",
"Definition:Triangular Number"
] | [
"Definition:Fibonacci Number",
"Odd Square is Eight Triangles Plus One",
"Definition:Triangular Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-13017 | Repdigit Triangular Numbers | The only repdigit numbers which are also triangular are:
:$55, 66, 666$
{{OEIS|A045914}} | {{begin-eqn}}
{{eqn | l = 55
| r = \dfrac {10 \times 11} 2
}}
{{eqn | l = 66
| r = \dfrac {11 \times 12} 2
}}
{{eqn | l = 666
| r = \dfrac {36 \times 37} 2
}}
{{end-eqn}}
Let $\dfrac {d \paren {10^j - 1}} 9$ be a $j$-digit repdigit number.
Suppose it is triangular.
For $d = 1$, write:
{{begin-eqn}}
{{... | The only [[Definition:Repdigit Number|repdigit numbers]] which are also [[Definition:Triangular Number|triangular]] are:
:$55, 66, 666$
{{OEIS|A045914}} | {{begin-eqn}}
{{eqn | l = 55
| r = \dfrac {10 \times 11} 2
}}
{{eqn | l = 66
| r = \dfrac {11 \times 12} 2
}}
{{eqn | l = 666
| r = \dfrac {36 \times 37} 2
}}
{{end-eqn}}
Let $\dfrac {d \paren {10^j - 1}} 9$ be a $j$-[[Definition:Digit|digit]] [[Definition:Repdigit Number|repdigit number]].
Suppose... | Repdigit Triangular Numbers | https://proofwiki.org/wiki/Repdigit_Triangular_Numbers | https://proofwiki.org/wiki/Repdigit_Triangular_Numbers | [
"Repdigit Numbers",
"Triangular Numbers"
] | [
"Definition:Repdigit Number",
"Definition:Triangular Number"
] | [
"Definition:Digit",
"Definition:Repdigit Number",
"Definition:Triangular Number",
"1 plus Power of 2 is not Perfect Power except 9",
"Definition:Triangular Number",
"Definition:Repunit",
"Odd Square is Eight Triangles Plus One",
"Definition:Square Number",
"Definition:Square Number",
"Square Modul... |
proofwiki-13018 | Increasing and Ordering on Mappings implies Mapping is Composition | Let $L = \struct {S, \preceq}, R = \struct {T, \preccurlyeq}$ be ordered sets.
Let $g: S \to T, d: T \to S$ be mappings such that
:$g$ and $d$ are increasing mappings
and
:$d \circ g \preceq I_S$ and $I_T \preccurlyeq g \circ d$
where $\preceq, \preccurlyeq$ denotes the orderings on mappings.
Then $d = d \circ \paren {... | Let $t \in T$.
By definition of ordering on mappings:
:$\map {I_T} t \preccurlyeq \map {\paren {g \circ d} } t$
By definition of identity mapping:
:$t \preccurlyeq \map {\paren {g \circ d} } t$
By definition of increasing mapping:
:$\map d t \preceq \map d {\map {\paren {g \circ d} } t}$
By definition of composition of... | Let $L = \struct {S, \preceq}, R = \struct {T, \preccurlyeq}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T, d: T \to S$ be [[Definition:Mapping|mappings]] such that
:$g$ and $d$ are [[Definition:Increasing Mapping|increasing mappings]]
and
:$d \circ g \preceq I_S$ and $I_T \preccurlyeq g \circ d$
where ... | Let $t \in T$.
By definition of [[Definition:Ordering on Mappings|ordering on mappings]]:
:$\map {I_T} t \preccurlyeq \map {\paren {g \circ d} } t$
By definition of [[Definition:Identity Mapping|identity mapping]]:
:$t \preccurlyeq \map {\paren {g \circ d} } t$
By definition of [[Definition:Increasing Mapping|increa... | Increasing and Ordering on Mappings implies Mapping is Composition | https://proofwiki.org/wiki/Increasing_and_Ordering_on_Mappings_implies_Mapping_is_Composition | https://proofwiki.org/wiki/Increasing_and_Ordering_on_Mappings_implies_Mapping_is_Composition | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Increasing/Mapping",
"Definition:Ordering on Mappings"
] | [
"Definition:Ordering on Mappings",
"Definition:Identity Mapping",
"Definition:Increasing/Mapping",
"Definition:Composition of Mappings",
"Definition:Ordering on Mappings",
"Definition:Identity Mapping",
"Definition:Composition of Mappings",
"Composition of Mappings is Associative",
"Definition:Antis... |
proofwiki-13019 | Closed Form for Square Pyramidal Numbers | The closed-form expression for the $n$th square pyramidal number is:
:$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{k \mathop = 1}^n k^2
| c = {{Defof|Square Pyramidal Number}}
}}
{{eqn | r = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6
| c = Sum of Sequence of Squares
}}
{{end-eqn}}
{{qed}} | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Square Pyramidal Number|square pyramidal number]] is:
:$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ | {{begin-eqn}}
{{eqn | l = S_n
| r = \sum_{k \mathop = 1}^n k^2
| c = {{Defof|Square Pyramidal Number}}
}}
{{eqn | r = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6
| c = [[Sum of Sequence of Squares]]
}}
{{end-eqn}}
{{qed}} | Closed Form for Square Pyramidal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Square_Pyramidal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Square_Pyramidal_Numbers | [
"Closed Forms",
"Pyramidal Numbers"
] | [
"Definition:Closed Form Expression",
"Definition:Square Pyramidal Number"
] | [
"Sum of Sequence of Squares"
] |
proofwiki-13020 | Closed Form for Pentagonal Pyramidal Numbers | The closed-form expression for the $n$th pentagonal pyramidal number is:
:$Q_n = \dfrac {n^2 \paren {n + 1} } 2$ | {{begin-eqn}}
{{eqn | l = Q_n
| r = \sum_{k \mathop = 1}^n P_k
| c = {{Defof|Pentagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {3 k - 1} } 2
| c = Closed Form for Pentagonal Numbers
}}
{{eqn | r = \dfrac 1 2 \paren {3 \sum_{k \mathop = 1}^n k^2 - \sum_{k \mathop = 1}... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Pentagonal Pyramidal Number|pentagonal pyramidal number]] is:
:$Q_n = \dfrac {n^2 \paren {n + 1} } 2$ | {{begin-eqn}}
{{eqn | l = Q_n
| r = \sum_{k \mathop = 1}^n P_k
| c = {{Defof|Pentagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {3 k - 1} } 2
| c = [[Closed Form for Pentagonal Numbers]]
}}
{{eqn | r = \dfrac 1 2 \paren {3 \sum_{k \mathop = 1}^n k^2 - \sum_{k \mathop ... | Closed Form for Pentagonal Pyramidal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentagonal_Pyramidal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentagonal_Pyramidal_Numbers | [
"Closed Forms",
"Pyramidal Numbers"
] | [
"Definition:Closed Form Expression",
"Definition:Pentagonal Pyramidal Number"
] | [
"Closed Form for Pentagonal Numbers",
"Sum of Sequence of Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-13021 | Surjection from Class to Proper Class | Let $A$ be a class.
Let $\mathrm P$ be a proper class.
Let $f: A \to \mathrm P$ be a surjection.
Then $A$ is proper. | {{AimForCont}} $A$ is not proper.
Then $A$ must be a set.
By the Axiom of Powers, $\powerset A$ is also a set.
Let $g: f \sqbrk A \to \powerset A$ be defined as:
:$\map g {\map f a} = f^{-1} \sqbrk {\set {\map f a} }$
It should be noted that:
:$\forall a \in A: \map {\paren {g \circ f} } a \ne \O$:
Suppose that $\map g... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\mathrm P$ be a [[Definition:Proper Class|proper class]].
Let $f: A \to \mathrm P$ be a [[Definition:Class Surjection|surjection]].
Then $A$ is [[Definition:Proper Class|proper]]. | {{AimForCont}} $A$ is not [[Definition:Proper Class|proper]].
Then $A$ must be a [[Definition:Set|set]].
By the [[Axiom:Axiom of Powers (Class Theory)|Axiom of Powers]], $\powerset A$ is also a [[Definition:Set|set]].
Let $g: f \sqbrk A \to \powerset A$ be defined as:
:$\map g {\map f a} = f^{-1} \sqbrk {\set {\map... | Surjection from Class to Proper Class | https://proofwiki.org/wiki/Surjection_from_Class_to_Proper_Class | https://proofwiki.org/wiki/Surjection_from_Class_to_Proper_Class | [
"Gödel-Bernays Class Theory",
"Class Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Surjection/Class Theory",
"Definition:Class (Class Theory)/Proper Class"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Set",
"Axiom:Axiom of Powers/Class Theory",
"Definition:Set",
"Definition:Singleton",
"Diagonal Relation is Equivalence",
"Definition:Injection/Class Theory",
"Definition:Surjection/Class Theory",
"Definition:Surjection",
"Injection from ... |
proofwiki-13022 | Image of Set under Mapping is Set | Let $A$ be a class.
Let $\mathrm U$ denote the universal class.
Let $f: A \to \mathrm U$ be a class mapping.
Let $S$ be a subset of $A$.
{{explain|Note that further work is needed on the Subset page to clarify that a "subset" of a "proper class" is indeed a "set". Some words are already on that page, but this needs to ... | {{NotZFC}}
{{AimForCont}} $f \sqbrk S$ is not a set.
Then $f \sqbrk S$ must be proper.
By Restriction of Mapping to Image is Surjection, the restriction $f \restriction_{S \times f \sqbrk S}$ is a surjection.
But this contradicts Surjection from Class to Proper Class.
Thus by contradiction, $f \sqbrk S$ is a set.
Hence... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\mathrm U$ denote the [[Definition:Universal Class|universal class]].
Let $f: A \to \mathrm U$ be a [[Definition:Class Mapping|class mapping]].
Let $S$ be a [[Definition:Subset|subset]] of $A$.
{{explain|Note that further work is needed on the Subset page... | {{NotZFC}}
{{AimForCont}} $f \sqbrk S$ is not a set.
Then $f \sqbrk S$ must be [[Definition:Proper Class|proper]].
By [[Restriction of Mapping to Image is Surjection]], the [[Definition:Restriction of Mapping|restriction]] $f \restriction_{S \times f \sqbrk S}$ is a [[Definition:Class Surjection|surjection]].
But... | Image of Set under Mapping is Set | https://proofwiki.org/wiki/Image_of_Set_under_Mapping_is_Set | https://proofwiki.org/wiki/Image_of_Set_under_Mapping_is_Set | [
"Gödel-Bernays Class Theory",
"Class Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Universal Class",
"Definition:Mapping/Class Theory",
"Definition:Subset",
"Definition:Mapping/Class Theory",
"Definition:Set",
"Definition:Set",
"Axiom:Axiom of Replacement",
"Definition:Zermelo-Fraenkel Set Theory"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Restriction of Mapping to Image is Surjection",
"Definition:Restriction/Mapping",
"Definition:Surjection/Class Theory",
"Surjection from Class to Proper Class",
"Definition:Contradiction",
"Definition:Set",
"Category:Gödel-Bernays Class Theory",
"Cate... |
proofwiki-13023 | Square Pyramidal and Triangular Numbers | The only positive integers which are simultaneously square pyramidal and triangular are:
:$1, 55, 91, 208 \, 335$
{{OEIS|A039596}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6
| c = Closed Form for Square Pyramidal Numbers
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 55
| r = \dfrac {5 \paren {5 ... | The only [[Definition:Positive Integer|positive integers]] which are simultaneously [[Definition:Square Pyramidal Number|square pyramidal]] and [[Definition:Triangular Number|triangular]] are:
:$1, 55, 91, 208 \, 335$
{{OEIS|A039596}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6
| c = [[Closed Form for Square Pyramidal Numbers]]
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 55
| r = \dfrac {5 ... | Square Pyramidal and Triangular Numbers | https://proofwiki.org/wiki/Square_Pyramidal_and_Triangular_Numbers | https://proofwiki.org/wiki/Square_Pyramidal_and_Triangular_Numbers | [
"Triangular Numbers",
"Pyramidal Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Square Pyramidal Number",
"Definition:Triangular Number"
] | [
"Closed Form for Square Pyramidal Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Square Pyramidal Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Square Pyramidal Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Square Pyramidal Numbers",
"Closed Form... |
proofwiki-13024 | Palindromes Formed by Multiplying by 55 | $55$ multiplied by any of the odd integers between $91$ and $109$ inclusive produces a palindromic number. | We have that:
:$55 \times 91 = 5005$
Then we have that:
:$55 \times 2 = 110$
Thus:
:$\forall k: 1 \le k \le 9: 55 \times 2 k = 110k = 110, 220, 330, \ldots, 990$
Thus for $1 \le k \le 9$:
:$55 \times \paren {91 + 2 k} = \sqbrk {5kk5}_{10}$
That is:
{{begin-eqn}}
{{eqn | l = 55 \times 91
| r = 5005
}}
{{eqn | l = ... | $55$ [[Definition:Integer Multiplication|multiplied]] by any of the [[Definition:Odd Integer|odd integers]] between $91$ and $109$ inclusive produces a [[Definition:Palindromic Number|palindromic number]]. | We have that:
:$55 \times 91 = 5005$
Then we have that:
:$55 \times 2 = 110$
Thus:
:$\forall k: 1 \le k \le 9: 55 \times 2 k = 110k = 110, 220, 330, \ldots, 990$
Thus for $1 \le k \le 9$:
:$55 \times \paren {91 + 2 k} = \sqbrk {5kk5}_{10}$
That is:
{{begin-eqn}}
{{eqn | l = 55 \times 91
| r = 5005
}}
{{eqn... | Palindromes Formed by Multiplying by 55 | https://proofwiki.org/wiki/Palindromes_Formed_by_Multiplying_by_55 | https://proofwiki.org/wiki/Palindromes_Formed_by_Multiplying_by_55 | [
"Recreational Mathematics",
"55"
] | [
"Definition:Multiplication/Integers",
"Definition:Odd Integer",
"Definition:Palindromic Number"
] | [] |
proofwiki-13025 | Dirichlet Convolution is Commutative | :$f * g = g * f$ | From the definition of the Dirichlet convolution:
:$\ds \map {\paren {f * g} } n = \sum_{a b \mathop = n} \map f a \map g b$
By definition, arithmetic functions are mappings from the natural numbers $\N$ to the complex numbers $\C$.
Thus $\map f a, \map g b \in \C$ and commutativity follows from Complex Multiplication ... | :$f * g = g * f$ | From the definition of the [[Definition:Dirichlet Convolution|Dirichlet convolution]]:
:$\ds \map {\paren {f * g} } n = \sum_{a b \mathop = n} \map f a \map g b$
By definition, [[Definition:Arithmetic Function|arithmetic functions]] are mappings from the [[Definition:Natural Numbers|natural numbers]] $\N$ to the [[De... | Dirichlet Convolution is Commutative | https://proofwiki.org/wiki/Dirichlet_Convolution_is_Commutative | https://proofwiki.org/wiki/Dirichlet_Convolution_is_Commutative | [
"Dirichlet Convolution"
] | [] | [
"Definition:Dirichlet Convolution",
"Definition:Arithmetic Function",
"Definition:Natural Numbers",
"Definition:Complex Number",
"Definition:Commutative/Operation",
"Complex Multiplication is Commutative"
] |
proofwiki-13026 | Dirichlet Convolution is Associative | : $\paren {f * g} * h = f * \paren {g * h}$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\paren {f * g} * h} } n
| r = \sum_{a b \mathop = n} \map {\paren {f * g} } a \map h b
}}
{{eqn | r = \sum_{a b \mathop = n} \ \sum_{c d \mathop = a} \map f c \map g d \map h b
}}
{{eqn | r = \sum_{b c d \mathop = n} \map f c \map g d \map h b
}}
{{end-eqn}}
and
{{... | : $\paren {f * g} * h = f * \paren {g * h}$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\paren {f * g} * h} } n
| r = \sum_{a b \mathop = n} \map {\paren {f * g} } a \map h b
}}
{{eqn | r = \sum_{a b \mathop = n} \ \sum_{c d \mathop = a} \map f c \map g d \map h b
}}
{{eqn | r = \sum_{b c d \mathop = n} \map f c \map g d \map h b
}}
{{end-eqn}}
and
... | Dirichlet Convolution is Associative | https://proofwiki.org/wiki/Dirichlet_Convolution_is_Associative | https://proofwiki.org/wiki/Dirichlet_Convolution_is_Associative | [
"Dirichlet Convolution"
] | [] | [] |
proofwiki-13027 | Identity Element for Dirichlet Convolution | :$\iota * f = f$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\iota * f} } n
| r = \sum_{d \mathop \divides n} \delta_{d 1} \map f {\frac n d}
}}
{{eqn | r = \map f n
}}
{{end-eqn}}
Hence the result.
{{qed}} | :$\iota * f = f$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\iota * f} } n
| r = \sum_{d \mathop \divides n} \delta_{d 1} \map f {\frac n d}
}}
{{eqn | r = \map f n
}}
{{end-eqn}}
Hence the result.
{{qed}} | Identity Element for Dirichlet Convolution | https://proofwiki.org/wiki/Identity_Element_for_Dirichlet_Convolution | https://proofwiki.org/wiki/Identity_Element_for_Dirichlet_Convolution | [
"Dirichlet Convolution"
] | [] | [] |
proofwiki-13028 | Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element | Let $L = \struct {S, \preceq}, R = \paren {T, \precsim}$ be ordered sets.
Let $g: S \to T, d:T \to S$ be mappings such that:
:$\tuple {g, d}$ is a Galois connection
and
:$g$ is a surjection.
Then
:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$ | By definition of Galois connection:
:$g$ is an increasing mapping.
Let $t \in T$.
By definition of surjection:
:$\Img g = T$
By {{Corollary|Image of Preimage under Mapping}}:
:$g \sqbrk {g^{-1} \sqbrk {t^\succeq} } = t^\succeq$
By Galois Connection is Expressed by Minimum:
:$\map d t = \min \set {g^{-1} \sqbrk {t^\succ... | Let $L = \struct {S, \preceq}, R = \paren {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T, d:T \to S$ be [[Definition:Mapping|mappings]] such that:
:$\tuple {g, d}$ is a [[Definition:Galois Connection|Galois connection]]
and
:$g$ is a [[Definition:Surjection|surjection]].
Then
:$\forall t ... | By definition of [[Definition:Galois Connection|Galois connection]]:
:$g$ is an [[Definition:Increasing Mapping|increasing mapping]].
Let $t \in T$.
By definition of [[Definition:Surjection|surjection]]:
:$\Img g = T$
By {{Corollary|Image of Preimage under Mapping}}:
:$g \sqbrk {g^{-1} \sqbrk {t^\succeq} } = t^\succ... | Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element | https://proofwiki.org/wiki/Upper_Adjoint_of_Galois_Connection_is_Surjection_implies_Lower_Adjoint_at_Element_is_Minimum_of_Preimage_of_Singleton_of_Element | https://proofwiki.org/wiki/Upper_Adjoint_of_Galois_Connection_is_Surjection_implies_Lower_Adjoint_at_Element_is_Minimum_of_Preimage_of_Singleton_of_Element | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Surjection"
] | [
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Surjection",
"Galois Connection is Expressed by Minimum",
"Definition:Min Operation",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Upper Closure/Element",
"Definition:Smallest Element",
"Definition:Infimum of... |
proofwiki-13029 | Dirichlet Convolution Preserves Multiplicativity | Let $f, g: \N \to \C$ be multiplicative arithmetic functions.
Then their Dirichlet convolution $f * g$ is again multiplicative. | Let $m, n$ be coprime integers.
By definition of multiplicative functions, we have:
:$(1): \quad \map f {m n} = \map f m \map f n$
:$(2): \quad \map g {m n} = \map g m \map g n$
{{begin-eqn}}
{{eqn | l = \map {\paren {f * g} } {m n}
| r = \sum_{d \mathop \divides m n} \map f d \map g {\frac {m n} d}
| c = {... | Let $f, g: \N \to \C$ be [[Definition:Multiplicative Arithmetic Function|multiplicative arithmetic functions]].
Then their [[Definition:Dirichlet Convolution|Dirichlet convolution]] $f * g$ is again [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | Let $m, n$ be [[Definition:Coprime Integers|coprime integers]].
By definition of [[Definition:Multiplicative Arithmetic Function|multiplicative functions]], we have:
:$(1): \quad \map f {m n} = \map f m \map f n$
:$(2): \quad \map g {m n} = \map g m \map g n$
{{begin-eqn}}
{{eqn | l = \map {\paren {f * g} } {m n}
... | Dirichlet Convolution Preserves Multiplicativity | https://proofwiki.org/wiki/Dirichlet_Convolution_Preserves_Multiplicativity | https://proofwiki.org/wiki/Dirichlet_Convolution_Preserves_Multiplicativity | [
"Dirichlet Convolution",
"Multiplicative Functions"
] | [
"Definition:Multiplicative Arithmetic Function",
"Definition:Dirichlet Convolution",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Coprime/Integers",
"Definition:Multiplicative Arithmetic Function",
"Divisors of Product of Coprime Integers",
"Definition:Multiplicative Arithmetic Function"
] |
proofwiki-13030 | Dirichlet Convolution Preserves Multiplicativity/General Result | Let $S \subset \N$ be a set of natural numbers with the property:
:$m n \in S, \map \gcd {m, n} = 1 \implies m, n \in S$
Define:
:$\map {\paren {f*_S g} } n = \ds \sum_{\substack {d \mathop \divides n \\ d \mathop \in S} } \map f d \map g {n / d}$
Then $f*_S g$ is multiplicative. | {{ProofWanted}}
Category:Dirichlet Convolution
Category:Multiplicative Functions
6pxmrgwltwzdbylg0m216tm3dnlo927 | Let $S \subset \N$ be a [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] with the property:
:$m n \in S, \map \gcd {m, n} = 1 \implies m, n \in S$
Define:
:$\map {\paren {f*_S g} } n = \ds \sum_{\substack {d \mathop \divides n \\ d \mathop \in S} } \map f d \map g {n / d}$
Then $f*_S g$ is [[... | {{ProofWanted}}
[[Category:Dirichlet Convolution]]
[[Category:Multiplicative Functions]]
6pxmrgwltwzdbylg0m216tm3dnlo927 | Dirichlet Convolution Preserves Multiplicativity/General Result | https://proofwiki.org/wiki/Dirichlet_Convolution_Preserves_Multiplicativity/General_Result | https://proofwiki.org/wiki/Dirichlet_Convolution_Preserves_Multiplicativity/General_Result | [
"Dirichlet Convolution",
"Multiplicative Functions"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Multiplicative Arithmetic Function"
] | [
"Category:Dirichlet Convolution",
"Category:Multiplicative Functions"
] |
proofwiki-13031 | Even Integer with Abundancy Index greater than 9 | Let $n \in \Z_{>0}$ have an abundancy index greater than $9$.
Then $n$ has at least $35$ distinct prime factors. | As Divisor Sum Function is Multiplicative, it follows easily that abundancy index is multiplicative as well.
We have for any prime $p$ and positive integer $k$:
{{begin-eqn}}
{{eqn | l = \frac {\map {\sigma_1} {p^k} } {p^k}
| r = \frac {p^{k + 1} - 1} {p^k \paren {p - 1} }
| c = Divisor Sum of Power of Prim... | Let $n \in \Z_{>0}$ have an [[Definition:Abundancy Index|abundancy index]] greater than $9$.
Then $n$ has at least $35$ [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]]. | As [[Divisor Sum Function is Multiplicative]], it follows easily that [[Definition:Abundancy Index|abundancy index]] is [[Definition:Multiplicative Arithmetic Function|multiplicative]] as well.
We have for any [[Definition:Prime Number|prime]] $p$ and [[Definition:Positive Integer|positive integer]] $k$:
{{begin-eqn}}... | Even Integer with Abundancy Index greater than 9 | https://proofwiki.org/wiki/Even_Integer_with_Abundancy_Index_greater_than_9 | https://proofwiki.org/wiki/Even_Integer_with_Abundancy_Index_greater_than_9 | [
"Abundancy"
] | [
"Definition:Abundancy Index",
"Definition:Distinct",
"Definition:Prime Factor"
] | [
"Divisor Sum Function is Multiplicative",
"Definition:Abundancy Index",
"Definition:Multiplicative Arithmetic Function",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Divisor Sum of Power of Prime",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Abundancy Index",
"Definition:P... |
proofwiki-13032 | Closed Form for Pentatope Numbers | The closed-form expression for the $n$th pentatope number is:
:$P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$ | {{begin-eqn}}
{{eqn | l = P_n
| r = \sum_{k \mathop = 1}^n T_k
| c = {{Defof|Pentatope Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \frac {n \paren {n + 1} \paren {n + 2} } 6
| c = Closed Form for Tetrahedral Numbers
}}
{{eqn | r = \sum_{k \mathop = 1}^n \frac {\paren {n^3 + 3 n^2 + 2 n} } 6
|... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Pentatope Number|pentatope number]] is:
:$P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$ | {{begin-eqn}}
{{eqn | l = P_n
| r = \sum_{k \mathop = 1}^n T_k
| c = {{Defof|Pentatope Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \frac {n \paren {n + 1} \paren {n + 2} } 6
| c = [[Closed Form for Tetrahedral Numbers]]
}}
{{eqn | r = \sum_{k \mathop = 1}^n \frac {\paren {n^3 + 3 n^2 + 2 n} } 6
... | Closed Form for Pentatope Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentatope_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Pentatope_Numbers | [
"Closed Forms",
"Pentatope Numbers"
] | [
"Definition:Closed Form Expression",
"Definition:Pentatope Number"
] | [
"Closed Form for Tetrahedral Numbers",
"Summation is Linear",
"Closed Form for Triangular Numbers",
"Sum of Sequence of Squares",
"Sum of Sequence of Cubes",
"Definition:Fraction/Denominator"
] |
proofwiki-13033 | Relation Between Rank and Exponent of Convergence | Let $f: \C \to \C$ be an entire function.
Let $k$ be its rank and $\tau$ be its exponent of convergence.
Then:
* $k=\tau=0$ if $f$ has finitely many zeroes.
* $k<\tau\leq k+1$ otherwise. | {{ProofWanted}}
Category:Entire Functions
epim9a757ti02kdsc835c98mkdrn4my | Let $f: \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $k$ be its [[Definition:Rank of Entire Function|rank]] and $\tau$ be its [[Definition:Exponent of Convergence|exponent of convergence]].
Then:
* $k=\tau=0$ if $f$ has [[Definition:Finitely Many|finitely many]] zeroes.
* $k<\tau\leq k+1$ oth... | {{ProofWanted}}
[[Category:Entire Functions]]
epim9a757ti02kdsc835c98mkdrn4my | Relation Between Rank and Exponent of Convergence | https://proofwiki.org/wiki/Relation_Between_Rank_and_Exponent_of_Convergence | https://proofwiki.org/wiki/Relation_Between_Rank_and_Exponent_of_Convergence | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Rank of Entire Function",
"Definition:Exponent of Convergence",
"Definition:Finite Set"
] | [
"Category:Entire Functions"
] |
proofwiki-13034 | Necessary and Sufficient Condition for First Order System to be Field for Second Order System | Let $\mathbf y$, $\mathbf f$, $\boldsymbol \psi$ be N-dimensional vectors.
Let $\boldsymbol\psi$ be continuously differentiable.
Then $\forall x \in \closedint a b$ the first-order system of differential equations:
:$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$
is a field for the second-order system
:$\mathbf y... | === Necessary condition ===
Differentiate the first-order system {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = \mathbf y''
| r = \frac {\d \boldsymbol \psi} {\d x}
}}
{{eqn | r = \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \frac... | Let $\mathbf y$, $\mathbf f$, $\boldsymbol \psi$ be [[Definition:Dimension |N-dimensional]] [[Definition:Vector|vectors]].
Let $\boldsymbol\psi$ be [[Definition:Continuously Differentiable Vector|continuously differentiable]].
Then $\forall x \in \closedint a b$ the [[Definition:First Order Ordinary Differential Equ... | === Necessary condition ===
[[Definition:Derivative|Differentiate]] the [[Definition:First Order Ordinary Differential Equation|first-order]] [[Definition:System of Differential Equations|system]] {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = \mathbf y''
| r = \frac {\d \boldsymbol \psi} {\d x}
}}
{{eq... | Necessary and Sufficient Condition for First Order System to be Field for Second Order System | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Field_for_Second_Order_System | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Field_for_Second_Order_System | [
"Calculus of Variations"
] | [
"Definition:Dimension ",
"Definition:Vector",
"Definition:Continuously Differentiable Vector",
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/System",
"Definition:Field of Directions",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Dif... | [
"Definition:Derivative",
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/System",
"Definition:Differential Equation/System",
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/System",
"Definition:Second Order Ordinary Di... |
proofwiki-13035 | Tetrahedral Number as Sum of Squares | :$H_n = \ds \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$
where $H_n$ denotes the $n$th tetrahedral number. | Let $n$ be even such that $n = 2 m$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2
| r = \sum_{k \mathop = 0}^m \paren {2 m - 2 k}^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^m \paren {2 k}^2
| c = Permutation of Indices of Summation
}}
{{eqn | r = \frac {2 m \par... | :$H_n = \ds \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$
where $H_n$ denotes the $n$th [[Definition:Tetrahedral Number|tetrahedral number]]. | Let $n$ be [[Definition:Even Integer|even]] such that $n = 2 m$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2
| r = \sum_{k \mathop = 0}^m \paren {2 m - 2 k}^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^m \paren {2 k}^2
| c = [[Permutation of Indices of Summation... | Tetrahedral Number as Sum of Squares | https://proofwiki.org/wiki/Tetrahedral_Number_as_Sum_of_Squares | https://proofwiki.org/wiki/Tetrahedral_Number_as_Sum_of_Squares | [
"Tetrahedral Number as Sum of Squares",
"Tetrahedral Numbers",
"Square Numbers"
] | [
"Definition:Tetrahedral Number"
] | [
"Definition:Even Integer",
"Permutation of Indices of Summation",
"Sum of Sequence of Even Squares",
"Closed Form for Tetrahedral Numbers",
"Definition:Odd Integer",
"Permutation of Indices of Summation",
"Closed Form for Tetrahedral Numbers",
"Category:Tetrahedral Number as Sum of Squares",
"Catego... |
proofwiki-13036 | Sum of Sequence of Odd Squares/Corollary | :$\ds \forall n \in \N: \sum_{r \mathop = 0}^n \paren {2 r + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^n \paren {2 n + 1}^2
| r = \sum_{r \mathop = 0}^{n - 1} \paren {2 r + 1}^2 + \paren {2 n + 1}^2
| c =
}}
{{eqn | r = \sum_{r \mathop = 1}^n \paren {2 \paren {r - 1} + 1}^2 + \paren {2 n + 1}^2
| c = Translation of Index Variable of Summation
}}
{{eqn | r... | :$\ds \forall n \in \N: \sum_{r \mathop = 0}^n \paren {2 r + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^n \paren {2 n + 1}^2
| r = \sum_{r \mathop = 0}^{n - 1} \paren {2 r + 1}^2 + \paren {2 n + 1}^2
| c =
}}
{{eqn | r = \sum_{r \mathop = 1}^n \paren {2 \paren {r - 1} + 1}^2 + \paren {2 n + 1}^2
| c = [[Translation of Index Variable of Summation]]
}}
{{eqn... | Sum of Sequence of Odd Squares/Corollary/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Squares/Corollary | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Squares/Corollary/Proof_1 | [
"Sum of Sequence of Odd Squares"
] | [] | [
"Translation of Index Variable of Summation",
"Sum of Sequence of Odd Squares",
"Difference of Two Squares"
] |
proofwiki-13037 | Sum of Sequence of Odd Squares/Corollary | :$\ds \forall n \in \N: \sum_{r \mathop = 0}^n \paren {2 r + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^n \paren {2 r + 1}^2
| r = \sum_{r \mathop = 0}^n \paren {2 r}^2 + \sum_{r \mathop = 0}^n 4 i + \sum_{r \mathop = 0}^n 1
| c =
}}
{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{r \mathop = 0}^n r + \sum_{r \mathop = 0}^n 1
| c = Sum... | :$\ds \forall n \in \N: \sum_{r \mathop = 0}^n \paren {2 r + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^n \paren {2 r + 1}^2
| r = \sum_{r \mathop = 0}^n \paren {2 r}^2 + \sum_{r \mathop = 0}^n 4 i + \sum_{r \mathop = 0}^n 1
| c =
}}
{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3 + 4 \sum_{r \mathop = 0}^n r + \sum_{r \mathop = 0}^n 1
| c = [[S... | Sum of Sequence of Odd Squares/Corollary/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Squares/Corollary | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Squares/Corollary/Proof_2 | [
"Sum of Sequence of Odd Squares"
] | [] | [
"Sum of Sequence of Even Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-13038 | Sum of Sequence of Even Squares | :$\ds \forall n \in \N: \sum_{i \mathop = 0}^n \paren {2 i}^2 = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 0}^n \paren {2 i}^2
| r = 4 \sum_{i \mathop = 1}^n i^2
| c = adjustment of indices: $4 i^2 = 0$ when $i = 0$
}}
{{eqn | r = 4 \frac {n \paren {n + 1} \paren {2 n + 1} } 6
| c = Sum of Sequence of Squares
}}
{{eqn | r = \frac {2 n \paren {n + 1} \paren {2 n +... | :$\ds \forall n \in \N: \sum_{i \mathop = 0}^n \paren {2 i}^2 = \frac {2 n \paren {n + 1} \paren {2 n + 1} } 3$ | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 0}^n \paren {2 i}^2
| r = 4 \sum_{i \mathop = 1}^n i^2
| c = adjustment of indices: $4 i^2 = 0$ when $i = 0$
}}
{{eqn | r = 4 \frac {n \paren {n + 1} \paren {2 n + 1} } 6
| c = [[Sum of Sequence of Squares]]
}}
{{eqn | r = \frac {2 n \paren {n + 1} \paren {2... | Sum of Sequence of Even Squares | https://proofwiki.org/wiki/Sum_of_Sequence_of_Even_Squares | https://proofwiki.org/wiki/Sum_of_Sequence_of_Even_Squares | [
"Square Numbers",
"Sums of Sequences"
] | [] | [
"Sum of Sequence of Squares",
"Category:Square Numbers",
"Category:Sums of Sequences"
] |
proofwiki-13039 | Killing Form of Orthogonal Lie Algebra | Let $\mathbb K \in \set {\C, \R}$.
Let $n$ be a positive integer.
Let $\map {\mathfrak {so}_n} {\mathbb K}$ be the Lie algebra of the special orthogonal group $\map {\operatorname {SO_n} } {\mathbb K}$.
Then its Killing form is $B: \tuple {X, Y} \mapsto \paren {n - 2} \map \tr {X Y}$. | === Lemma ===
Let $R$ be a ring with unity.
Let $n$ be a positive integer.
Let $E_{ij}$ denote the matrix with only zeroes except a $1$ at the $\tuple {i, j}$th position.
Then for all $X, Y \in R^{n \times n}$:
:$\ds \sum_{1 \mathop \le i \mathop < j \le n} \map \tr {\paren {\map X {E_{ij} - E_{ji} } Y}^t \paren {E_{ij... | Let $\mathbb K \in \set {\C, \R}$.
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $\map {\mathfrak {so}_n} {\mathbb K}$ be the [[Definition:Lie Algebra|Lie algebra]] of the [[Definition:Special Orthogonal Group|special orthogonal group]] $\map {\operatorname {SO_n} } {\mathbb K}$.
Then its [[Def... | === Lemma ===
Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $E_{ij}$ denote the [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $\tuple {i, j}$th position.
Then for all $X, Y \in R^{n \times n}$:
:$\ds \sum_{1 \math... | Killing Form of Orthogonal Lie Algebra | https://proofwiki.org/wiki/Killing_Form_of_Orthogonal_Lie_Algebra | https://proofwiki.org/wiki/Killing_Form_of_Orthogonal_Lie_Algebra | [
"Lie Algebras"
] | [
"Definition:Positive/Integer",
"Definition:Lie Algebra",
"Definition:Special Orthogonal Group",
"Definition:Killing Form"
] | [
"Definition:Ring with Unity",
"Definition:Positive/Integer",
"Definition:Matrix",
"Trace of Alternating Product of Matrices and Almost Zero Matrices",
"Definition:Frobenius Inner Product",
"Trace in Terms of Orthonormal Basis",
"Definition:Orthonormal Basis"
] |
proofwiki-13040 | Killing Form of Symplectic Lie Algebra | Let $\mathbb K \in \set {\C, \R}$.
Let $n$ be a positive integer.
Let $\map {\mathfrak {sp}_{2 n} } {\mathbb K}$ be the Lie algebra of the symplectic group $\map {\operatorname {Sp} } {2 n, \mathbb K}$.
Then its Killing form is $B: \tuple {X, Y} \mapsto \paren {2 n + 2} \map \tr {X Y}$. | === Lemma ===
Let $R$ be a ring with unity.
Let $n$ be a positive integer.
Let $E_{ij}$ denote the matrix with only zeroes except a $1$ at the $\tuple {i, j}$th position.
Let $X, Y \in R^{2 n \times 2 n}$.
Let $X = \begin {pmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{pmatrix}$ and $Y = \begin{pmatrix} Y_{11} & Y_{1... | Let $\mathbb K \in \set {\C, \R}$.
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $\map {\mathfrak {sp}_{2 n} } {\mathbb K}$ be the [[Definition:Lie Algebra|Lie algebra]] of the [[Definition:Symplectic Group|symplectic group]] $\map {\operatorname {Sp} } {2 n, \mathbb K}$.
Then its [[Definition:... | === Lemma ===
Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $E_{ij}$ denote the [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $\tuple {i, j}$th position.
Let $X, Y \in R^{2 n \times 2 n}$.
Let $X = \begin {pmatrix... | Killing Form of Symplectic Lie Algebra | https://proofwiki.org/wiki/Killing_Form_of_Symplectic_Lie_Algebra | https://proofwiki.org/wiki/Killing_Form_of_Symplectic_Lie_Algebra | [
"Lie Algebras"
] | [
"Definition:Positive/Integer",
"Definition:Lie Algebra",
"Definition:Symplectic Group",
"Definition:Killing Form"
] | [
"Definition:Ring with Unity",
"Definition:Positive/Integer",
"Definition:Matrix"
] |
proofwiki-13041 | Trace of Alternating Product of Matrices and Almost Zero Matrices | Let $R$ be a ring with unity.
Let $n, m$ be positive integers.
Let $E_{ij}$ denote the $n \times n$ matrix with only zeroes except a $1$ at the $\tuple {i, j}$th element.
Let $A_1, \ldots, A_m \in R^{n \times n}$.
Let $i_k, j_k \in \set {1, \ldots, n}$ for $k \in \set {1, \ldots, m}$.
Let $i_0 = i_m$ and $j_0 = j_m$.
T... | Use induction and the facts $E_{i j} A E_{k l} = A_{j k} E_{i l}$ and $\map \tr {A E_{i j} } = A_{j i}$ (induction basis).
{{ProofWanted}}
Category:Matrix Theory
90xzg6kakcyz1uk8egkxzr0imns4wuo | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $n, m$ be [[Definition:Positive Integer|positive integers]].
Let $E_{ij}$ denote the $n \times n$ [[Definition:Matrix|matrix]] with only zeroes except a $1$ at the $\tuple {i, j}$th [[Definition:Element of Matrix|element]].
Let $A_1, \ldots, A_m \in R^... | Use induction and the facts $E_{i j} A E_{k l} = A_{j k} E_{i l}$ and $\map \tr {A E_{i j} } = A_{j i}$ (induction basis).
{{ProofWanted}}
[[Category:Matrix Theory]]
90xzg6kakcyz1uk8egkxzr0imns4wuo | Trace of Alternating Product of Matrices and Almost Zero Matrices | https://proofwiki.org/wiki/Trace_of_Alternating_Product_of_Matrices_and_Almost_Zero_Matrices | https://proofwiki.org/wiki/Trace_of_Alternating_Product_of_Matrices_and_Almost_Zero_Matrices | [
"Matrix Theory"
] | [
"Definition:Ring with Unity",
"Definition:Positive/Integer",
"Definition:Matrix",
"Definition:Matrix/Element"
] | [
"Category:Matrix Theory"
] |
proofwiki-13042 | Trace in Terms of Orthonormal Basis | Let $\mathbb K \subset \C$ be a field.
{{explain|Can the field be more general than just being a subfield of $\C$? Can it not in fact be an arbitrary Definition:Valued Field?}}
Let $\struct {V, \innerprod {\, \cdot \,} {\, \cdot \,} }$ be an inner product space over $\mathbb K$ of dimension $n$.
Let $\tuple {e_1, \ldot... | Let $\ds \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
Let $A$ be the matrix relative to the basis $\tuple {e_1, \ldots, e_n}$.
Then by the above assumption, $A_{ij} = c_{ij}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \tr f
| r = \map \tr A
| c = {{Defof|Trace of Linear Operator}}
}}
{{eqn | r = \sum_{i \ma... | Let $\mathbb K \subset \C$ be a [[Definition:Field (Abstract Algebra)|field]].
{{explain|Can the field be more general than just being a subfield of $\C$? Can it not in fact be an arbitrary [[Definition:Valued Field]]?}}
Let $\struct {V, \innerprod {\, \cdot \,} {\, \cdot \,} }$ be an [[Definition:Inner Product Space... | Let $\ds \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
Let $A$ be the [[Definition:Matrix|matrix]] [[Definition:Relative Matrix of Linear Transformation|relative]] to the basis $\tuple {e_1, \ldots, e_n}$.
Then by the above assumption, $A_{ij} = c_{ij}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \tr f
| r = \... | Trace in Terms of Orthonormal Basis | https://proofwiki.org/wiki/Trace_in_Terms_of_Orthonormal_Basis | https://proofwiki.org/wiki/Trace_in_Terms_of_Orthonormal_Basis | [
"Linear Algebra"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Valued Field",
"Definition:Inner Product Space",
"Definition:Dimension of Vector Space",
"Definition:Orthonormal Basis",
"Definition:Linear Operator",
"Definition:Trace (Linear Algebra)/Linear Operator"
] | [
"Definition:Matrix",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Inner Product"
] |
proofwiki-13043 | Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic | Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be ordered sets.
Ley $g:S \to T, d:T \to S$ be mappings such that
:$\left({g, d}\right)$ is Galois connection
and
:$g$ is a surjection/
Let $N = \left({d\left[{T}\right], \preceq'}\right)$ be an ordered subset of $L$.
Then $R$ and $N$ are order isomorp... | By Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection:
:$d$ is an injection.
Define $d' = d:T \to g\left[{T}\right]$
By definition:
:$d'$ is an injection.
By definition of Galois connection:
:$d$ is an increasing mapping.
We will prove that
:$d$ is order embedding.
Let $x, y \in T$.
By... | Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be [[Definition:Ordered Set|ordered sets]].
Ley $g:S \to T, d:T \to S$ be [[Definition:Mapping|mappings]] such that
:$\left({g, d}\right)$ is [[Definition:Galois Connection|Galois connection]]
and
:$g$ is a [[Definition:Surjection|surjection]]/
Let $... | By [[Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection]]:
:$d$ is an [[Definition:Injection|injection]].
Define $d' = d:T \to g\left[{T}\right]$
By definition:
:$d'$ is an [[Definition:Injection|injection]].
By definition of [[Definition:Galois Connection|Galois connection]]:
:$d$ ... | Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic | https://proofwiki.org/wiki/Galois_Connection_with_Upper_Adjoint_Surjective_implies_Scond_Ordered_Set_and_Image_of_Lower_Adjoint_are_Isomorphic | https://proofwiki.org/wiki/Galois_Connection_with_Upper_Adjoint_Surjective_implies_Scond_Ordered_Set_and_Image_of_Lower_Adjoint_are_Isomorphic | [
"Galois Connections",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Surjection",
"Definition:Ordered Subset",
"Definition:Order Isomorphism"
] | [
"Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Definition:Order Embedding",
"Definition:Increasing/Mapping",
"Definition:Ordered Subset",
"Upper ... |
proofwiki-13044 | Trace in Terms of Dual Basis | Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of dimension $n$.
Let $\tuple {e_1, \ldots, e_n}$ be a basis of $M$.
Let $\tuple {e_1^*,\ldots, e_n^*}$ be its dual basis
Let $f: M \to M$ be a linear operator.
Then its trace equals:
:$\map \tr f = \ds \sum_{i \mathop = 1}^n e_i^* \paren {\map f {e_i} }$ | Let $\ds \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
Let $A$ be the matrix relative to the basis $\tuple {e_1, \ldots, e_n}$.
Then by the above assumption:
:$A_{ij} = c_{ij}$
Then:
{{begin-eqn}}
{{eqn | l = \map \tr f
| r = \map \tr A
| c = {{Defof|Trace of Linear Operator}}
}}
{{eqn | r = \sum_{i \ma... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] of [[Definition:Dimension of Module|dimension]] $n$.
Let $\tuple {e_1, \ldots, e_n}$ be a [[Definition:Basis of Module|basis]] of $M$.
Let $\tuple {e_1^*,\ldots, e_n^*}$ be its [[Definition:... | Let $\ds \map f {e_i} = \sum_{j \mathop = 1}^n c_{ij} e_j$
Let $A$ be the [[Definition:Matrix|matrix]] [[Definition:Relative Matrix of Linear Transformation|relative]] to the basis $\tuple {e_1, \ldots, e_n}$.
Then by the above assumption:
:$A_{ij} = c_{ij}$
Then:
{{begin-eqn}}
{{eqn | l = \map \tr f
| r = \... | Trace in Terms of Dual Basis | https://proofwiki.org/wiki/Trace_in_Terms_of_Dual_Basis | https://proofwiki.org/wiki/Trace_in_Terms_of_Dual_Basis | [
"Linear Algebra"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension of Module",
"Definition:Basis of Module",
"Definition:Ordered Dual Basis",
"Definition:Linear Operator",
"Definition:Trace (Linear Algebra)/Linear Operator"
] | [
"Definition:Matrix",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Linear Form (Linear Algebra)"
] |
proofwiki-13045 | Internal Angle of Equilateral Triangle | The internal angles of an equilateral triangle measure $60^\circ$ or $\dfrac \pi 3$ radians. | By definition, an equilateral triangle is a regular polygon with $3$ sides.
From Internal Angles of Regular Polygon, the size $A$ of each internal angle of a regular $n$-gon is given by:
:$A = \dfrac {\paren {n - 2} 180^\circ} n$
Thus:
:$A = \dfrac {180^\circ} n = 60^\circ$
From Value of Degree in Radians:
:$1^\circ = ... | The [[Definition:Internal Angle|internal angles]] of an [[Definition:Equilateral Triangle|equilateral triangle]] measure $60^\circ$ or $\dfrac \pi 3$ [[Definition:Radian|radians]]. | By definition, an [[Definition:Equilateral Triangle|equilateral triangle]] is a [[Definition:Regular Polygon|regular polygon]] with $3$ [[Definition:Side of Polygon|sides]].
From [[Internal Angles of Regular Polygon]], the size $A$ of each [[Definition:Internal Angle|internal angle]] of a [[Definition:Regular Polygon|... | Internal Angle of Equilateral Triangle | https://proofwiki.org/wiki/Internal_Angle_of_Equilateral_Triangle | https://proofwiki.org/wiki/Internal_Angle_of_Equilateral_Triangle | [
"Equilateral Triangles"
] | [
"Definition:Polygon/Internal Angle",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Angular Measure/Radian"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Internal Angles of Regular Polygon",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Regular",
"Value of Degree in Radians"
] |
proofwiki-13046 | Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways | The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$. | {{begin-eqn}}
{{eqn | l = 60
| r = 7 + 53
| c = $6$ ways
}}
{{eqn | r = 13 + 47
| c =
}}
{{eqn | r = 17 + 43
| c =
}}
{{eqn | r = 19 + 41
| c =
}}
{{eqn | r = 23 + 37
| c =
}}
{{eqn | r = 29 + 31
| c =
}}
{{end-eqn}}
It is determined that there are no smaller numbers with ... | The smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $6$ different ways is $60$. | {{begin-eqn}}
{{eqn | l = 60
| r = 7 + 53
| c = $6$ ways
}}
{{eqn | r = 13 + 47
| c =
}}
{{eqn | r = 17 + 43
| c =
}}
{{eqn | r = 19 + 41
| c =
}}
{{eqn | r = 23 + 37
| c =
}}
{{eqn | r = 29 + 31
| c =
}}
{{end-eqn}}
It is determined that there are no smaller numbers wit... | Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Odd_Primes_in_6_Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Odd_Primes_in_6_Ways | [
"Prime Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Odd Prime"
] | [
"Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Odd Prime"
] |
proofwiki-13047 | Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways | The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows:
:{| border="1"
|-
! align="right" style = "padding: 2px 10px" | $k$
! align="right" style = "padding: 2px 10px" | $n$
|-
| align="right" style = "padding: 2px 10px" | $1$
... | {{begin-eqn}}
{{eqn | l = 6
| r = 3 + 3
| c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8
| r = 5 + 3
| c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 10
| r = 7 + 3
| c = $2$ ways
}}
{{eqn | r = 5 + 5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 12
| r = ... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] $n$ which are the smallest such that they are the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Prime|odd primes]] in $k$ different ways begins as follows:
:{| border="1"
|-
! align="right" style = "padding:... | {{begin-eqn}}
{{eqn | l = 6
| r = 3 + 3
| c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8
| r = 5 + 3
| c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 10
| r = 7 + 3
| c = $2$ ways
}}
{{eqn | r = 5 + 5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 12
... | Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Odd_Primes_in_n_Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Odd_Primes_in_n_Ways | [
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Odd Prime"
] | [
"Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Odd Prime"
] |
proofwiki-13048 | Change of Basis Matrix from Basis to Itself is Identity | Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of finite dimension $n > 0$.
Let $\BB$ be an ordered basis of $M$.
Then the change of basis matrix from $\BB$ to $\BB$ is the $n\times n$ identity matrix:
:$\mathbf M_{\BB, \BB} = \mathbf I$ | Follows directly from the definition of change of basis matrix.
{{qed}}
Category:Change of Basis
d4nra8dzu15mpluuopjb97y6u2boq46 | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n > 0$.
Let $\BB$ be an [[Definition:Ordered Basis|ordered basis]] of $M$.
Then the [[Definition:Change of Basis Matrix|change... | Follows directly from the definition of [[Definition:Change of Basis Matrix|change of basis matrix]].
{{qed}}
[[Category:Change of Basis]]
d4nra8dzu15mpluuopjb97y6u2boq46 | Change of Basis Matrix from Basis to Itself is Identity | https://proofwiki.org/wiki/Change_of_Basis_Matrix_from_Basis_to_Itself_is_Identity | https://proofwiki.org/wiki/Change_of_Basis_Matrix_from_Basis_to_Itself_is_Identity | [
"Change of Basis"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Change of Basis Matrix",
"Definition:Unit Matrix"
] | [
"Definition:Change of Basis Matrix",
"Category:Change of Basis"
] |
proofwiki-13049 | Sum of Sequence of Seventh Powers | :$\ds \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^0 j^7
| r = 0
| c =
}}
{{eqn ... | :$\ds \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eq... | Sum of Sequence of Seventh Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Seventh_Powers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Seventh_Powers | [
"Sums of Sequences",
"Seventh Powers"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-13050 | Change of Coordinate Vector Under Change of Basis | Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of finite dimension $n > 0$.
Let $\BB$ and $\CC$ be bases of $M$.
Let $\mathbf M_{\BB, \CC}$ be the change of basis matrix from $\BB$ to $\CC$.
Let $m \in M$.
Let $\sqbrk m_\BB$ and $\sqbrk m_\CC$ be its coordinate vectors relative to $\BB$ and $\CC$ respective... | Intuitively, when we consider $\BB$ and $\CC$ as row vectors, this is because:
:$\CC = \BB \cdot \mathbf M_{\BB, \CC}$ and:
:$\BB \cdot \sqbrk m_\BB = \CC \cdot \sqbrk m_\CC$ imply:
:$\BB \cdot \sqbrk m_\BB = \BB \cdot \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.
Because $\BB$ is a basis, this implies $\sqbrk m_\BB = \ma... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n > 0$.
Let $\BB$ and $\CC$ be [[Definition:Basis of Module|bases]] of $M$.
Let $\mathbf M_{\BB, \CC}$ be the [[Definition:Chan... | Intuitively, when we consider $\BB$ and $\CC$ as row vectors, this is because:
:$\CC = \BB \cdot \mathbf M_{\BB, \CC}$ and:
:$\BB \cdot \sqbrk m_\BB = \CC \cdot \sqbrk m_\CC$ imply:
:$\BB \cdot \sqbrk m_\BB = \BB \cdot \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.
Because $\BB$ is a [[Definition:Basis of Module|basis]], t... | Change of Coordinate Vector Under Change of Basis | https://proofwiki.org/wiki/Change_of_Coordinate_Vector_Under_Change_of_Basis | https://proofwiki.org/wiki/Change_of_Coordinate_Vector_Under_Change_of_Basis | [
"Change of Basis"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Basis of Module",
"Definition:Change of Basis Matrix",
"Definition:Coordinate Vector"
] | [
"Definition:Basis of Module",
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Product",
"Definition:Associative Operation"
] |
proofwiki-13051 | Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint | Let $\mathbf p$ be continuously differentiable.
{{explain|What is $\mathbf p$, and differentiable with respect to what?}}
The boundary conditions
:$\bigvalueat {\map {\mathbf y} a} {x \mathop = a} = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$
are self-adjoint {{iff}}:
:$\forall i, k \in \N: 1 \l... | === Necessary Condition ===
By assumption the boundary conditions are self-adjoint.
Then there exists $\map g {x, \mathbf y}$ such that:
:$\map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } = \dfrac {\partial \map g {x \mathbf y} } {\partial y_i}$
Because $\mathbf p \in C^1$:
:$g \in C^2$
{{handwaving|Why?... | Let $\mathbf p$ be [[Definition:Continuously Differentiable|continuously differentiable]].
{{explain|What is $\mathbf p$, and differentiable with respect to what?}}
The [[Definition:Boundary Condition|boundary conditions]]
:$\bigvalueat {\map {\mathbf y} a} {x \mathop = a} = \bigvalueat {\map {\boldsymbol \psi} {\m... | === Necessary Condition ===
By assumption the [[Definition:Boundary Condition|boundary conditions]] are [[Definition:Self-Adjoint Boundary Conditions|self-adjoint]].
Then there exists $\map g {x, \mathbf y}$ such that:
:$\map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } = \dfrac {\partial \map g {x \ma... | Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Boundary_Conditions_to_be_Self-adjoint | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Boundary_Conditions_to_be_Self-adjoint | [
"Calculus of Variations"
] | [
"Definition:Continuously Differentiable",
"Definition:Boundary Condition",
"Definition:Self-Adjoint Boundary Conditions"
] | [
"Definition:Boundary Condition",
"Definition:Self-Adjoint Boundary Conditions",
"Definition:Derivative",
"Schwarz-Clairaut Theorem",
"Definition:Partial Derivative",
"Definition:Commutative/Elements",
"Definition:Boundary Condition",
"Definition:Self-Adjoint Boundary Conditions"
] |
proofwiki-13052 | Reciprocal of 61 | :$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$ | Performing the calculation using long division:
<pre>
0.01639344262295081967213114754098360655737704918032786885245901...
--------------------------------------------------------------------
61)1.00000000000000000000000000000000000000000000000000000000000000000
61 122 61 61 488 183 18... | :$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.01639344262295081967213114754098360655737704918032786885245901...
--------------------------------------------------------------------
61)1.00000000000000000000000000000000000000000000000000000000000000000
61 122 61 ... | Reciprocal of 61 | https://proofwiki.org/wiki/Reciprocal_of_61 | https://proofwiki.org/wiki/Reciprocal_of_61 | [
"61",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:61",
"Category:Examples of Reciprocals"
] |
proofwiki-13053 | Composition of Galois Connections is Galois Connection | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ and $L_3 = \struct {S_3, \preceq_3}$ be ordered sets.
Let:
:$g_1: S_1 \to S_2, g_2: S_2 \to S_3, d_1: S_2 \to S_1, d_2: S_3 \to S_2$
be mappings such that:
:$\tuple {g_1, d_1}$ and $\tuple {g_2, d_2}$
are Galois connections.
Then $\tuple {g_2 \circ ... | By definition of Galois connection:
:$g_1$, $g_2$, $d_2$, and $d_1$ are increasing mappings.
Thus by Composite of Increasing Mappings is Increasing:
:$g_2 \circ g_1$ and $d_1 \circ d_2$ are increasing mappings.
Let $s \in S_3, t \in S_1$.
We will prove that
:$s \preceq_3 \map {\paren {g_2 \circ g_1} } t \implies \map {... | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ and $L_3 = \struct {S_3, \preceq_3}$ be [[Definition:Ordered Set|ordered sets]].
Let:
:$g_1: S_1 \to S_2, g_2: S_2 \to S_3, d_1: S_2 \to S_1, d_2: S_3 \to S_2$
be [[Definition:Mapping|mappings]] such that:
:$\tuple {g_1, d_1}$ and $\tuple {g_2, d_2... | By definition of [[Definition:Galois Connection|Galois connection]]:
:$g_1$, $g_2$, $d_2$, and $d_1$ are [[Definition:Increasing Mapping|increasing mappings]].
Thus by [[Composite of Increasing Mappings is Increasing]]:
:$g_2 \circ g_1$ and $d_1 \circ d_2$ are [[Definition:Increasing Mapping|increasing mappings]].
Le... | Composition of Galois Connections is Galois Connection | https://proofwiki.org/wiki/Composition_of_Galois_Connections_is_Galois_Connection | https://proofwiki.org/wiki/Composition_of_Galois_Connections_is_Galois_Connection | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Galois Connection"
] | [
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Composite of Increasing Mappings is Increasing",
"Definition:Increasing/Mapping",
"Definition:Composition of Mappings",
"Definition:Galois Connection",
"Definition:Increasing/Mapping",
"Galois Connection Implies Order on Mappings",
"D... |
proofwiki-13054 | Vector Space has Basis | Let $K$ be a division ring.
Let $V$ be a vector space over $K$.
Then $V$ has a basis. | The result follows from Vector Space has Basis between Linearly Independent Set and Spanning Set.
It suffices to find a linearly independent subset $L \subseteq V$ that is contained in a spanning set $S \subseteq V$.
By Empty Set is Linearly Independent, $L$ can be taken to be the empty set.
Or if $V$ is nonzero, by Si... | Let $K$ be a [[Definition:Division Ring|division ring]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $K$.
Then $V$ has a [[Definition:Basis of Vector Space|basis]]. | The result follows from [[Vector Space has Basis between Linearly Independent Set and Spanning Set]].
It suffices to find a [[Definition:Linearly Independent Set|linearly independent]] [[Definition:Subset|subset]] $L \subseteq V$ that is [[Definition:Contain|contained]] in a [[Definition:Spanning Set of Vector Space|s... | Vector Space has Basis | https://proofwiki.org/wiki/Vector_Space_has_Basis | https://proofwiki.org/wiki/Vector_Space_has_Basis | [
"Bases of Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Basis of Vector Space"
] | [
"Vector Space has Basis between Linearly Independent Set and Spanning Set",
"Definition:Linearly Independent/Set",
"Definition:Subset",
"Definition:Subset",
"Definition:Generator of Vector Space",
"Empty Set is Linearly Independent",
"Definition:Empty Set",
"Singleton is Linearly Independent",
"Defi... |
proofwiki-13055 | Product of Change of Basis Matrices | Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of finite dimension $n>0$.
Let $\AA$, $\BB$ and $\CC$ be ordered bases of $M$.
Let $\mathbf M_{\AA, \BB}$, $\mathbf M_{\BB, \CC}$ and $\mathbf M_{\AA, \CC}$ be the change of basis matrices from $\AA$ to $\BB$, $\BB$ to $\CC$ and $\AA$ to $\CC$ respectively.
The... | Let $m \in M$.
Let $\sqbrk m_\AA$ be its coordinate vector relative to $\AA$, and similarly for $\BB$ and $\CC$.
On the one hand:
{{begin-eqn}}
{{eqn | l = \sqbrk m_\AA
| r = \mathbf M_{\AA, \CC} \cdot \sqbrk m_\CC
| c = Change of Coordinate Vector Under Change of Basis
}}
{{end-eqn}}
On the other hand:
{{b... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
Let $\AA$, $\BB$ and $\CC$ be [[Definition:Ordered Basis|ordered bases]] of $M$.
Let $\mathbf M_{\AA, \BB}$, $\mathbf M_{... | Let $m \in M$.
Let $\sqbrk m_\AA$ be its [[Definition:Coordinate Vector|coordinate vector]] relative to $\AA$, and similarly for $\BB$ and $\CC$.
On the one hand:
{{begin-eqn}}
{{eqn | l = \sqbrk m_\AA
| r = \mathbf M_{\AA, \CC} \cdot \sqbrk m_\CC
| c = [[Change of Coordinate Vector Under Change of Basi... | Product of Change of Basis Matrices | https://proofwiki.org/wiki/Product_of_Change_of_Basis_Matrices | https://proofwiki.org/wiki/Product_of_Change_of_Basis_Matrices | [
"Change of Basis"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Change of Basis Matrix"
] | [
"Definition:Coordinate Vector",
"Change of Coordinate Vector Under Change of Basis",
"Change of Coordinate Vector Under Change of Basis",
"Change of Coordinate Vector Under Change of Basis"
] |
proofwiki-13056 | Relative Matrix of Composition of Linear Transformations | Let $R$ be a ring with unity.
Let $M, N, P$ be free $R$-modules of finite dimension $m, n, p > 0$ respectively.
Let $\AA, \BB, \CC$ be ordered bases of $M, N, P$.
Let $f: M \to N$ and $g : N \to P$ be linear transformations, and $g \circ f$ be their composition.
Let $\mathbf M_{f, \BB, \AA}$ and $\mathbf M_{g, \CC, \BB... | Let $m \in M$, and $\sqbrk m_\AA$ be its coordinate vector with respect to $\AA$.
On the one hand:
{{begin-eqn}}
{{eqn | l = \sqbrk {\map g {\map f m} }_\CC
| r = \mathbf M_{g \mathop \circ f, \CC, \AA} \cdot \sqbrk m_\AA
| c = Change of Coordinate Vectors Under Linear Mapping applied to $g \circ f$
}}
{{en... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M, N, P$ be [[Definition:Free Module over Ring|free $R$-modules]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $m, n, p > 0$ respectively.
Let $\AA, \BB, \CC$ be [[Definition:Ordered Basis (Linear Algebra)|ordered bases]] of $M, N, P$.... | Let $m \in M$, and $\sqbrk m_\AA$ be its [[Definition:Coordinate Vector|coordinate vector]] with respect to $\AA$.
On the one hand:
{{begin-eqn}}
{{eqn | l = \sqbrk {\map g {\map f m} }_\CC
| r = \mathbf M_{g \mathop \circ f, \CC, \AA} \cdot \sqbrk m_\AA
| c = [[Change of Coordinate Vectors Under Linear ... | Relative Matrix of Composition of Linear Transformations/Proof 1 | https://proofwiki.org/wiki/Relative_Matrix_of_Composition_of_Linear_Transformations | https://proofwiki.org/wiki/Relative_Matrix_of_Composition_of_Linear_Transformations/Proof_1 | [
"Relative Matrices of Linear Transformations",
"Relative Matrix of Composition of Linear Transformations"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Linear Transformation",
"Definition:Composition of Mappings",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Matrix",
"Definition:Rel... | [
"Definition:Coordinate Vector",
"Change of Coordinate Vectors Under Linear Transformation",
"Change of Coordinate Vectors Under Linear Transformation",
"Change of Coordinate Vectors Under Linear Transformation"
] |
proofwiki-13057 | Relative Matrix of Composition of Linear Transformations | Let $R$ be a ring with unity.
Let $M, N, P$ be free $R$-modules of finite dimension $m, n, p > 0$ respectively.
Let $\AA, \BB, \CC$ be ordered bases of $M, N, P$.
Let $f: M \to N$ and $g : N \to P$ be linear transformations, and $g \circ f$ be their composition.
Let $\mathbf M_{f, \BB, \AA}$ and $\mathbf M_{g, \CC, \BB... | Let:
:$\AA = \sequence {a_m}$
:$\BB = \sequence {b_n}$
:$\CC = \sequence {c_p}$
Let:
:$\sqbrk \alpha_{m n} = \sqbrk {f; \sequence {b_n}, \sequence {a_m} }$
and:
:$\sqbrk \beta_{n p} = \sqbrk {g; \sequence {c_p}, \sequence {b_n} }$
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {g \circ f} } {a_j}
| r = \map g {\map... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M, N, P$ be [[Definition:Free Module over Ring|free $R$-modules]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $m, n, p > 0$ respectively.
Let $\AA, \BB, \CC$ be [[Definition:Ordered Basis (Linear Algebra)|ordered bases]] of $M, N, P$.... | Let:
:$\AA = \sequence {a_m}$
:$\BB = \sequence {b_n}$
:$\CC = \sequence {c_p}$
Let:
:$\sqbrk \alpha_{m n} = \sqbrk {f; \sequence {b_n}, \sequence {a_m} }$
and:
:$\sqbrk \beta_{n p} = \sqbrk {g; \sequence {c_p}, \sequence {b_n} }$
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {g \circ f} } {a_j}
| r = \map g ... | Relative Matrix of Composition of Linear Transformations/Proof 2 | https://proofwiki.org/wiki/Relative_Matrix_of_Composition_of_Linear_Transformations | https://proofwiki.org/wiki/Relative_Matrix_of_Composition_of_Linear_Transformations/Proof_2 | [
"Relative Matrices of Linear Transformations",
"Relative Matrix of Composition of Linear Transformations"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Linear Transformation",
"Definition:Composition of Mappings",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Matrix",
"Definition:Rel... | [] |
proofwiki-13058 | Change of Coordinate Vectors Under Linear Transformation | Let $R$ be a ring with unity.
Let $M, N$ be free $R$-modules of finite dimension $m, n > 0$ respectively.
Let $\AA, \BB$ be ordered bases of $M$ and $N$ respectively.
Let $f: M \to N$ be a linear transformation.
Let $\mathbf M_{f, \BB, \AA}$ be its matrix relative to $\AA$ and $\BB$.
Then for all $m \in M$:
:$\sqbrk {\... | Both sides are linear in $m$ and they coincide on the elements of $\AA$ by definition of $\mathbf M_{f, \BB, \AA}$.
So they are equal for all $m \in M$.
{{explain|this has to be fleshed out}}
{{qed}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M, N$ be [[Definition:Free Module over Ring|free $R$-modules]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $m, n > 0$ respectively.
Let $\AA, \BB$ be [[Definition:Ordered Basis (Linear Algebra)|ordered bases]] of $M$ and $N$ respectiv... | Both sides are linear in $m$ and they coincide on the elements of $\AA$ by definition of $\mathbf M_{f, \BB, \AA}$.
So they are equal for all $m \in M$.
{{explain|this has to be fleshed out}}
{{qed}} | Change of Coordinate Vectors Under Linear Transformation | https://proofwiki.org/wiki/Change_of_Coordinate_Vectors_Under_Linear_Transformation | https://proofwiki.org/wiki/Change_of_Coordinate_Vectors_Under_Linear_Transformation | [
"Linear Transformations"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Linear Transformation",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Coordinate Vector",
"Definition:Basis (Linear Algebra)"
] | [] |
proofwiki-13059 | Matrix of Bilinear Form Under Change of Basis | Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of finite dimension $n > 0$.
Let $\AA$ and $\BB$ be ordered bases of $M$.
Let $\mathbf M_{\AA, \BB}$ be the change of basis matrix from $\AA$ to $\BB$.
Let $f : M \times M \to R$ be a bilinear form.
Let $\mathbf M_{f, \AA}$ be its matrix relative to $\AA$.
Then... | Let $m \in M$, and let $\sqbrk m_\AA$ and $\sqbrk m_\BB$ denote its coordinate vectors relative to $\AA$ and $\BB$.
We have:
{{begin-eqn}}
{{eqn | l = \map f m
| r = \sqbrk m_\AA^\intercal \cdot \mathbf M_{f, \AA} \cdot \sqbrk m_\AA
}}
{{eqn | r = \paren {\mathbf M_{\AA, \BB} \cdot \sqbrk m_\BB}^\intercal \cdot \... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n > 0$.
Let $\AA$ and $\BB$ be [[Definition:Ordered Basis|ordered bases]] of $M$.
Let $\mathbf M_{\AA, \BB}$ be the [[Definitio... | Let $m \in M$, and let $\sqbrk m_\AA$ and $\sqbrk m_\BB$ denote its [[Definition:Coordinate Vector|coordinate vectors]] relative to $\AA$ and $\BB$.
We have:
{{begin-eqn}}
{{eqn | l = \map f m
| r = \sqbrk m_\AA^\intercal \cdot \mathbf M_{f, \AA} \cdot \sqbrk m_\AA
}}
{{eqn | r = \paren {\mathbf M_{\AA, \BB} \cd... | Matrix of Bilinear Form Under Change of Basis | https://proofwiki.org/wiki/Matrix_of_Bilinear_Form_Under_Change_of_Basis | https://proofwiki.org/wiki/Matrix_of_Bilinear_Form_Under_Change_of_Basis | [
"Bilinear Forms (Linear Algebra)",
"Change of Basis"
] | [
"Definition:Ring with Unity",
"Definition:Free Module over Ring",
"Definition:Dimension (Linear Algebra)",
"Definition:Ordered Basis",
"Definition:Change of Basis Matrix",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Relative Matrix of Bilinear Form",
"Definition:Relative Matrix of Bilinea... | [
"Definition:Coordinate Vector",
"Change of Coordinate Vector Under Change of Basis",
"Transpose of Matrix Product",
"Category:Bilinear Forms (Linear Algebra)",
"Category:Change of Basis"
] |
proofwiki-13060 | Symmetric Bilinear Form is Reflexive | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$.
Let $b$ be a bilinear form on $V$.
Let $b$ be symmetric.
Then $b$ is reflexive. | Let $\tuple {v, w} \in V \times V$ with $\map b {v, w} = 0$.
Because $b$ is symmetric, $\map b {w, v} = 0$.
Because $\tuple {v, w}$ was arbitrary, $b$ is reflexive.
{{qed}} | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
Let $b$ be a [[Definition:Bilinear Form (Linear Algebra)|bilinear form]] on $V$.
Let $b$ be [[Definition:Symmetric Bilinear Form|symmetric]].
Then $b$ is [[Definition:Reflexiv... | Let $\tuple {v, w} \in V \times V$ with $\map b {v, w} = 0$.
Because $b$ is [[Definition:Symmetric Bilinear Form|symmetric]], $\map b {w, v} = 0$.
Because $\tuple {v, w}$ was arbitrary, $b$ is [[Definition:Reflexive Bilinear Form|reflexive]].
{{qed}} | Symmetric Bilinear Form is Reflexive | https://proofwiki.org/wiki/Symmetric_Bilinear_Form_is_Reflexive | https://proofwiki.org/wiki/Symmetric_Bilinear_Form_is_Reflexive | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Symmetric Bilinear Form",
"Definition:Reflexive Bilinear Form"
] | [
"Definition:Symmetric Bilinear Form",
"Definition:Reflexive Bilinear Form"
] |
proofwiki-13061 | Alternating Bilinear Form is Reflexive | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$.
Let $b$ be a bilinear form on $V$.
Let $b$ be alternating.
Then $b$ is reflexive. | Let $\tuple {v, w} \in V \times V$ with $\map b {v, w} = 0$.
We have:
{{begin-eqn}}
{{eqn | l = 0
| r = \map b {v + w, v + w}
| c = $b$ is alternating
}}
{{eqn | r = \map b {v, v} + \map b {v, w} + \map b {w, v} + \map b {w, w}
| c = {{Defof|Bilinear Form (Linear Algebra)}}
}}
{{eqn | r = \map b {v, w... | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
Let $b$ be a [[Definition:Bilinear Form (Linear Algebra)|bilinear form]] on $V$.
Let $b$ be [[Definition:Alternating Bilinear Form|alternating]].
Then $b$ is [[Definition:Refl... | Let $\tuple {v, w} \in V \times V$ with $\map b {v, w} = 0$.
We have:
{{begin-eqn}}
{{eqn | l = 0
| r = \map b {v + w, v + w}
| c = $b$ is [[Definition:Alternating Bilinear Form|alternating]]
}}
{{eqn | r = \map b {v, v} + \map b {v, w} + \map b {w, v} + \map b {w, w}
| c = {{Defof|Bilinear Form (Li... | Alternating Bilinear Form is Reflexive | https://proofwiki.org/wiki/Alternating_Bilinear_Form_is_Reflexive | https://proofwiki.org/wiki/Alternating_Bilinear_Form_is_Reflexive | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Alternating Bilinear Form",
"Definition:Reflexive Bilinear Form"
] | [
"Definition:Alternating Bilinear Form",
"Definition:Alternating Bilinear Form",
"Definition:Reflexive Bilinear Form"
] |
proofwiki-13062 | Bilinear Form is Reflexive iff Symmetric or Alternating | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$.
Let $b$ be a bilinear form on $V$.
Then the following are equivalent:
:$(1): \quad$ $b$ is reflexive
:$(2): \quad$ $b$ is symmetric or alternating | === 1 implies 2 ===
Follows from Reflexive Bilinear Form is Symmetric or Alternating
{{qed}} | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
Let $b$ be a [[Definition:Bilinear Form (Linear Algebra)|bilinear form]] on $V$.
Then the following are [[Definition:Logically Equivalent|equivalent]]:
:$(1): \quad$ $b$ is [[... | === 1 implies 2 ===
Follows from [[Reflexive Bilinear Form is Symmetric or Alternating]]
{{qed}} | Bilinear Form is Reflexive iff Symmetric or Alternating | https://proofwiki.org/wiki/Bilinear_Form_is_Reflexive_iff_Symmetric_or_Alternating | https://proofwiki.org/wiki/Bilinear_Form_is_Reflexive_iff_Symmetric_or_Alternating | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Logical Equivalence",
"Definition:Reflexive Bilinear Form",
"Definition:Symmetric Bilinear Form",
"Definition:Alternating Bilinear Form"
] | [
"Reflexive Bilinear Form is Symmetric or Alternating"
] |
proofwiki-13063 | Reflexive Bilinear Form is Symmetric or Alternating | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$.
Let $f$ be a bilinear form on $V$.
Let $f$ be reflexive.
Then $f$ is symmetric or alternating. | Let $x, y, z \in V$.
Because $f$ is bilinear:
:$\map f {x, \map f {x, y} z - \map f {x, z} y} = \map f {x, y} \, \map f {x, z} - \map f {x, z} \, \map f {x, y} = 0$
Because $f$ is reflexive:
:$\map f {\map f {x, y} z - \map f {x, z} y, x} = 0$
Because $f$ is bilinear:
:$(1): \quad \map f {x, y} \, \map f {z, x} = \map ... | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$.
Let $f$ be a [[Definition:Bilinear Form (Linear Algebra)|bilinear form]] on $V$.
Let $f$ be [[Definition:Reflexive Bilinear Form|reflexive]].
Then $f$ is [[Definition:Symmetri... | Let $x, y, z \in V$.
Because $f$ is [[Definition:Bilinear Form (Linear Algebra)|bilinear]]:
:$\map f {x, \map f {x, y} z - \map f {x, z} y} = \map f {x, y} \, \map f {x, z} - \map f {x, z} \, \map f {x, y} = 0$
Because $f$ is [[Definition:Reflexive Bilinear Form|reflexive]]:
:$\map f {\map f {x, y} z - \map f {x, z}... | Reflexive Bilinear Form is Symmetric or Alternating | https://proofwiki.org/wiki/Reflexive_Bilinear_Form_is_Symmetric_or_Alternating | https://proofwiki.org/wiki/Reflexive_Bilinear_Form_is_Symmetric_or_Alternating | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Reflexive Bilinear Form",
"Definition:Symmetric Bilinear Form",
"Definition:Alternating Bilinear Form"
] | [
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Reflexive Bilinear Form",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Symmetric Bilinear Form",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Contradiction",
"Definition:Altern... |
proofwiki-13064 | Apéry's Theorem | Apéry's constant:
:$\map \zeta 3 = \ds \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$
is irrational. | We have:
:$\dfrac 6 {\map \zeta 3} = 5 - \cfrac {1^6} {117 - \cfrac {2^6} {535 - \cfrac {\dotsb} {\dotsb - \cfrac {n^6} {34 n^3 + 51 n^2 + 27 n + 5} - \dotsb } } }$
{{ProofWanted}} | [[Definition:Apéry's Constant|Apéry's constant]]:
:$\map \zeta 3 = \ds \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$
is [[Definition:Irrational Number|irrational]]. | We have:
:$\dfrac 6 {\map \zeta 3} = 5 - \cfrac {1^6} {117 - \cfrac {2^6} {535 - \cfrac {\dotsb} {\dotsb - \cfrac {n^6} {34 n^3 + 51 n^2 + 27 n + 5} - \dotsb } } }$
{{ProofWanted}} | Apéry's Theorem | https://proofwiki.org/wiki/Apéry's_Theorem | https://proofwiki.org/wiki/Apéry's_Theorem | [
"Apéry's Constant"
] | [
"Definition:Apéry's Constant",
"Definition:Irrational Number"
] | [] |
proofwiki-13065 | Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9 | The following positive integers $p$ have reciprocals whose decimal expansions:
:$(1): \quad$ have the maximum period, that is: $p - 1$
:$(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$, of each of the digits from $0$ to $9$:
::$61$, $131$, $\ldots$ | From Reciprocal of $61$:
{{:Reciprocal of 61}}
From Reciprocal of $131$:
{{:Reciprocal of 131}} | The following [[Definition:Positive Integer|positive integers]] $p$ have [[Definition:Reciprocal|reciprocals]] whose [[Definition:Decimal Expansion|decimal expansions]]:
:$(1): \quad$ have the maximum [[Definition:Period of Recurrence|period]], that is: $p - 1$
:$(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$,... | From [[Reciprocal of 61|Reciprocal of $61$]]:
{{:Reciprocal of 61}}
From [[Reciprocal of 131|Reciprocal of $131$]]:
{{:Reciprocal of 131}} | Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9 | https://proofwiki.org/wiki/Reciprocals_whose_Decimal_Expansion_contain_Equal_Numbers_of_Digits_from_0_to_9 | https://proofwiki.org/wiki/Reciprocals_whose_Decimal_Expansion_contain_Equal_Numbers_of_Digits_from_0_to_9 | [
"Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Reciprocal",
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Digit"
] | [
"Reciprocal of 61",
"Reciprocal of 131"
] |
proofwiki-13066 | Reciprocal of 131 | :$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$ | Performing the calculation using long division (zoom out for best effect):
<pre>
0.0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229
----------------------------------------------------------------------------------------------------... | :$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$ | Performing the calculation using long division (zoom out for best effect):
<pre>
0.0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229
---------------------------------------------------------------------------------------------------... | Reciprocal of 131 | https://proofwiki.org/wiki/Reciprocal_of_131 | https://proofwiki.org/wiki/Reciprocal_of_131 | [
"131",
"Examples of Reciprocals"
] | [] | [
"Category:131",
"Category:Examples of Reciprocals"
] |
proofwiki-13067 | Sequences of 4 Consecutive Integers with Rising Divisor Sum | The following ordered quadruples of consecutive integers have divisor sum values which are strictly increasing:
:$61, 62, 63, 64$
:$73, 74, 75, 76$ | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {61}
| r = 62
| c = Divisor Sum of Prime Number: $61$ is prime
}}
{{eqn | l = \map {\sigma_1} {62}
| r = 96
| c = {{DSFLink|62}}
}}
{{eqn | l = \map {\sigma_1} {63}
| r = 104
| c = {{DSFLink|63}}
}}
{{eqn | l = \map {\sigma_1} {64}
| r ... | The following [[Definition:Ordered Quadruple|ordered quadruples]] of consecutive [[Definition:Integer|integers]] have [[Definition:Divisor Sum Function|divisor sum]] values which are [[Definition:Strictly Increasing Mapping|strictly increasing]]:
:$61, 62, 63, 64$
:$73, 74, 75, 76$ | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {61}
| r = 62
| c = [[Divisor Sum of Prime Number]]: $61$ is [[Definition:Prime Number|prime]]
}}
{{eqn | l = \map {\sigma_1} {62}
| r = 96
| c = {{DSFLink|62}}
}}
{{eqn | l = \map {\sigma_1} {63}
| r = 104
| c = {{DSFLink|63}}
}}
{{eqn | l =... | Sequences of 4 Consecutive Integers with Rising Divisor Sum | https://proofwiki.org/wiki/Sequences_of_4_Consecutive_Integers_with_Rising_Divisor_Sum | https://proofwiki.org/wiki/Sequences_of_4_Consecutive_Integers_with_Rising_Divisor_Sum | [
"Divisor Sum Function"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Integer",
"Definition:Divisor Sum Function",
"Definition:Strictly Increasing/Mapping"
] | [
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Prime Number"
] |
proofwiki-13068 | Symmetric Bilinear Form can be Diagonalized | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$.
Let $f$ be a symmetric bilinear form on $V$.
Then there exists an ordered basis for which the relative matrix of $f$ is diagonal. | {{ProofWanted}}
Category:Bilinear Forms (Linear Algebra)
e4gaxj5kousfaflt2jcse0v2tz33i05 | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n>0$.
Let $f$ be a [[Definition:Symmetric Bilinear Form|symmetric bilinear form]] on $V$.
Then there exists an [[D... | {{ProofWanted}}
[[Category:Bilinear Forms (Linear Algebra)]]
e4gaxj5kousfaflt2jcse0v2tz33i05 | Symmetric Bilinear Form can be Diagonalized | https://proofwiki.org/wiki/Symmetric_Bilinear_Form_can_be_Diagonalized | https://proofwiki.org/wiki/Symmetric_Bilinear_Form_can_be_Diagonalized | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Dimension (Linear Algebra)",
"Definition:Symmetric Bilinear Form",
"Definition:Ordered Basis",
"Definition:Relative Matrix of Bilinear Form",
"Definition:Diagonal Matrix"
] | [
"Category:Bilinear Forms (Linear Algebra)"
] |
proofwiki-13069 | Dimension of Radical of Bilinear Form | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$ of finite dimension $n > 0$.
Let $f$ be a bilinear form on $V$.
Let $\map \Rad V$ denote the radical of $V$.
Let $\map {\operatorname {rk} } f$ be the rank of $f$.
Then:
:$\map \dim {\map \Rad V} = n - \map {\operatorname {rk} } f$
where $\dim$ deno... | {{ProofWanted}}
Category:Bilinear Forms (Linear Algebra)
2yz8g3246cqb4qistt492aef52neaj3 | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]] $n > 0$.
Let $f$ be a [[Definition:Bilinear Form (Linear Algebra)|bilinear form]] on $V$.
Let $\map \Rad V$ denote t... | {{ProofWanted}}
[[Category:Bilinear Forms (Linear Algebra)]]
2yz8g3246cqb4qistt492aef52neaj3 | Dimension of Radical of Bilinear Form | https://proofwiki.org/wiki/Dimension_of_Radical_of_Bilinear_Form | https://proofwiki.org/wiki/Dimension_of_Radical_of_Bilinear_Form | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Dimension (Linear Algebra)",
"Definition:Bilinear Form (Linear Algebra)",
"Definition:Orthogonal (Bilinear Form)/Radical",
"Definition:Rank of Bilinear Form",
"Definition:Dimension (Linear Algebra)"
] | [
"Category:Bilinear Forms (Linear Algebra)"
] |
proofwiki-13070 | Dimension of Orthogonal Complement With Respect to Bilinear Form | Let $\mathbb K$ be a field.
Let $V$ be a vector space over $\mathbb K$ of finite dimension.
Let $f$ be a nondegenerate bilinear form on $V$.
Let $U\subset V$ be a subspace.
Let $U^\perp$ be its orthogonal complement.
Then:
:$\map \dim U + \map \dim U^\perp = \map \dim V$ | {{ProofWanted}}
Category:Bilinear Forms (Linear Algebra)
k4ago7gqoii36ny5o63id7iu434c2mv | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $\mathbb K$ of [[Definition:Dimension (Linear Algebra)|finite dimension]].
Let $f$ be a [[Definition:Nondegenerate Bilinear Form|nondegenerate bilinear form]] on $V$.
Let $U\subset V$ be a [... | {{ProofWanted}}
[[Category:Bilinear Forms (Linear Algebra)]]
k4ago7gqoii36ny5o63id7iu434c2mv | Dimension of Orthogonal Complement With Respect to Bilinear Form | https://proofwiki.org/wiki/Dimension_of_Orthogonal_Complement_With_Respect_to_Bilinear_Form | https://proofwiki.org/wiki/Dimension_of_Orthogonal_Complement_With_Respect_to_Bilinear_Form | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Dimension (Linear Algebra)",
"Definition:Nondegenerate Bilinear Form",
"Definition:Vector Subspace",
"Definition:Orthogonal (Bilinear Form)/Orthogonal Complement"
] | [
"Category:Bilinear Forms (Linear Algebra)"
] |
proofwiki-13071 | Pluperfect Digital Invariant has less than 61 Digits | Let $n \in \Z_{>0}$ be a pluperfect digital invariant.
Then $n$ has less than $61$ digits. | We have that:
:$n \times 9^n < 10^\paren {n - 1}$
when $n > 60$.
So an $n$-digit integer, for $n > 60$, is always greater than the sum of the $n$th powers of its digits.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Pluperfect Digital Invariant|pluperfect digital invariant]].
Then $n$ has less than $61$ [[Definition:Digit|digits]]. | We have that:
:$n \times 9^n < 10^\paren {n - 1}$
when $n > 60$.
So an [[Definition:Digit|$n$-digit]] [[Definition:Positive Integer|integer]], for $n > 60$, is always greater than the [[Definition:Integer Addition|sum]] of the [[Definition:Integer Power|$n$th powers]] of its [[Definition:Digit|digits]].
{{qed}} | Pluperfect Digital Invariant has less than 61 Digits | https://proofwiki.org/wiki/Pluperfect_Digital_Invariant_has_less_than_61_Digits | https://proofwiki.org/wiki/Pluperfect_Digital_Invariant_has_less_than_61_Digits | [
"Pluperfect Digital Invariants"
] | [
"Definition:Pluperfect Digital Invariant",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Digit"
] |
proofwiki-13072 | Anisotropic Vector Gives Composition of Bilinear Space | Let $\mathbb K$ be a field.
Let $\struct {V, f}$ be a bilinear space over $\mathbb K$.
Let $v \in V$ be anisotropic.
Let $\sequence v$ be its span.
Let $v^\perp$ be its orthogonal complement.
Then $\struct {V, f}$ is the internal orthogonal sum of $\sequence v$ and $v^\perp$:
:$V = \sequence v \oplus v^\perp$ | {{ProofWanted}}
Category:Bilinear Forms (Linear Algebra)
6yx0osn0ure0y4izg9zk4dgq3191k38 | Let $\mathbb K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\struct {V, f}$ be a [[Definition:Bilinear Space|bilinear space]] over $\mathbb K$.
Let $v \in V$ be [[Definition:Anisotropic Vector (Bilinear Form)|anisotropic]].
Let $\sequence v$ be its [[Definition:Linear Span|span]].
Let $v^\perp$ be its ... | {{ProofWanted}}
[[Category:Bilinear Forms (Linear Algebra)]]
6yx0osn0ure0y4izg9zk4dgq3191k38 | Anisotropic Vector Gives Composition of Bilinear Space | https://proofwiki.org/wiki/Anisotropic_Vector_Gives_Composition_of_Bilinear_Space | https://proofwiki.org/wiki/Anisotropic_Vector_Gives_Composition_of_Bilinear_Space | [
"Bilinear Forms (Linear Algebra)"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Bilinear Space",
"Definition:Anisotropic Vector (Bilinear Form)",
"Definition:Generated Submodule/Linear Span",
"Definition:Orthogonal (Bilinear Form)/Orthogonal Complement",
"Definition:Internal Orthogonal Sum (Bilinear Space)"
] | [
"Category:Bilinear Forms (Linear Algebra)"
] |
proofwiki-13073 | Basis of Free Module is No Greater than Generator | Let $R$ be a commutative ring with unity.
Let $M$ be a free $R$-module with basis $B$.
Let $S$ be a generating set for $M$.
Then:
:$\size B \le \size S$.
That is, there exists an injection from $B$ to $S$. | Because $S$ is a generating set, there is a surjective homomorphism
:$\phi : R^{\paren S} \to M$
where $R^{\paren S}$ is the free $R$-module on $S$.
Because $B$ is a basis, there is an isomorphism
:$\psi : R^{\paren B} \to M$
Thus $f = \psi^{-1} \circ \phi: R^{\paren S} \to R^{\paren B}$ is a surjective module homomorp... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Free Module over Ring|free $R$-module]] with [[Definition:Basis (Linear Algebra)|basis]] $B$.
Let $S$ be a [[Definition:Generator of Module|generating set]] for $M$.
Then:
:$\size B \le \size S$.
That is... | Because $S$ is a [[Definition:Generator of Module|generating set]], there is a [[Definition:Surjection|surjective]] [[Definition:Module Homomorphism|homomorphism]]
:$\phi : R^{\paren S} \to M$
where $R^{\paren S}$ is the [[Definition:Free Module on Set|free $R$-module]] on $S$.
Because $B$ is a [[Definition:Basis of M... | Basis of Free Module is No Greater than Generator | https://proofwiki.org/wiki/Basis_of_Free_Module_is_No_Greater_than_Generator | https://proofwiki.org/wiki/Basis_of_Free_Module_is_No_Greater_than_Generator | [
"Free Modules"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Free Module over Ring",
"Definition:Basis (Linear Algebra)",
"Definition:Generator of Module",
"Definition:Injection"
] | [
"Definition:Generator of Module",
"Definition:Surjection",
"Definition:Linear Transformation",
"Definition:Free Module on Set",
"Definition:Basis of Module",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism",
"Definition:Surjection",
"Definition:Linear Tr... |
proofwiki-13074 | Extended Rolle's Theorem | Let $f: D \to \R$ be differentiable on a closed interval $I \subseteq \R$.
Let $x_0 < x_1 < \dots < x_n \in I$.
Let $\map f {x_i} = 0$ for $i = 0, \ldots, n$.
Then for all $i = 0, \ldots, n-1$:
:$\exists \xi_i \in \openint {x_i} {x_{i + 1} }: \map {f'} {\xi_i} = 0$ | Since $f$ is differentiable on $I$, f is differentiable on $\closedint {x_i} {x_{i + 1} }$ for $i = 0, \ldots, n - 1$.
Thus a fortiori, $f$ is also continuous on the closed interval $\closedint {x_i} {x_{i + 1} }$ and differentiable on the open interval $\openint {x_i} {x_{i + 1} }$.
For $i = 0, \ldots, n$ we have $\ma... | Let $f: D \to \R$ be [[Definition:Differentiable on Interval|differentiable]] on a [[Definition:Closed Real Interval|closed interval]] $I \subseteq \R$.
Let $x_0 < x_1 < \dots < x_n \in I$.
Let $\map f {x_i} = 0$ for $i = 0, \ldots, n$.
Then for all $i = 0, \ldots, n-1$:
:$\exists \xi_i \in \openint {x_i} {x_{i + 1... | Since $f$ is [[Definition:Differentiable on Interval|differentiable]] on $I$, f is [[Definition:Differentiable on Interval|differentiable]] on $\closedint {x_i} {x_{i + 1} }$ for $i = 0, \ldots, n - 1$.
Thus [[Definition:A Fortiori|a fortiori]], $f$ is also [[Definition:Continuous on Interval|continuous]] on the [[Def... | Extended Rolle's Theorem | https://proofwiki.org/wiki/Extended_Rolle's_Theorem | https://proofwiki.org/wiki/Extended_Rolle's_Theorem | [
"Differentiable Real Functions",
"Continuous Real Functions",
"Rolle's Theorem"
] | [
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Closed"
] | [
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:A Fortiori",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real... |
proofwiki-13075 | Pell's Equation/Examples/61 | :$x^2 - 61 y^2 = 1$
has the smallest positive integral solution:
:$x = 1 \, 766 \, 319 \, 049$
:$y = 226 \, 153 \, 980$ | From Continued Fraction Expansion of $\sqrt {61}$:
:$\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$
The cycle is of length is $11$.
By Solution of Pell's Equation, the only solutions of $x^2 - 61 y^2 = 1$ are:
:${p_{11 r} }^2 - 61 {q_{11 r} }^2 = \paren {-1}^{11 r}$
for $r = 1, 2, 3, \ldots$
Wh... | :$x^2 - 61 y^2 = 1$
has the smallest [[Definition:Positive Integer|positive integral]] solution:
:$x = 1 \, 766 \, 319 \, 049$
:$y = 226 \, 153 \, 980$ | From [[Continued Fraction Expansion of Irrational Square Root/Examples/61|Continued Fraction Expansion of $\sqrt {61}$]]:
:$\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fractio... | Pell's Equation/Examples/61 | https://proofwiki.org/wiki/Pell's_Equation/Examples/61 | https://proofwiki.org/wiki/Pell's_Equation/Examples/61 | [
"Pell's Equation",
"61"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/61",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents"
] |
proofwiki-13076 | No Infinitely Descending Membership Chains/Corollary | There cannot exist a sequence $\sequence {x_n}$ whose domain is $\N_{\gt 0}$ such that:
:$\forall n \in \N_{>0}: x_{n + 1} \in x_n$ | {{AimForCont}} there is a sequence like that.
From the definition of a sequence, let $f$ be the the mapping that is defined by $\sequence {x_n}$.
Let $\omega$ denote the minimally inductive set.
Let $g: \omega \to \N_{>0}$ be defined as:
:$\map g \alpha = \alpha + 1$
Then the composition $f \circ g$ is a mapping whose ... | There cannot exist a [[Definition:Sequence|sequence]] $\sequence {x_n}$ whose [[Definition:Domain of Mapping|domain]] is $\N_{\gt 0}$ such that:
:$\forall n \in \N_{>0}: x_{n + 1} \in x_n$ | {{AimForCont}} there is a sequence like that.
From the definition of a [[Definition:Sequence|sequence]], let $f$ be the the [[Definition:Mapping|mapping]] that is defined by $\sequence {x_n}$.
Let $\omega$ denote the [[Definition:Minimally Inductive Set|minimally inductive set]].
Let $g: \omega \to \N_{>0}$ be defi... | No Infinitely Descending Membership Chains/Corollary | https://proofwiki.org/wiki/No_Infinitely_Descending_Membership_Chains/Corollary | https://proofwiki.org/wiki/No_Infinitely_Descending_Membership_Chains/Corollary | [
"Axiom of Foundation"
] | [
"Definition:Sequence",
"Definition:Domain (Set Theory)/Mapping"
] | [
"Definition:Sequence",
"Definition:Mapping",
"Definition:Minimally Inductive Set",
"Definition:Composition of Mappings",
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Contradiction",
"No Infinitely Descending Membership Chains",
"Proof by Contradiction",
"Definition:S... |
proofwiki-13077 | Kaprekar's Process for 2-Digit Numbers | Kaprekar's process, when applied to a non-repdigit $2$-digit positive integer leads to the cycle:
:$09 \to 81 \to 63 \to 27 \to 45 \to 09$
Note that it is important to retain the leading zero on the $9$, or the process trivially terminates in $0$. | Let $n \in \Z_{>0}$ be a $2$-digit positive integer.
{{WLOG}}, let $n$ be expressed in decimal notation as:
:$n = 10 x + y$
where:
: $x > y$
:$0 \le x \le 9, 0 \le y \le 9$
The reversal of $n$ is $10 y + x$.
We have:
{{begin-eqn}}
{{eqn | l = 10x + y - 10 y - x
| r = 9 \left({x - y}\right)
| c = where $1 \l... | [[Definition:Kaprekar's Process|Kaprekar's process]], when applied to a non-[[Definition:Repdigit Number|repdigit]] [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integer]] leads to the cycle:
:$09 \to 81 \to 63 \to 27 \to 45 \to 09$
Note that it is important to retain the leading [[Definition:Z... | Let $n \in \Z_{>0}$ be a [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integer]].
{{WLOG}}, let $n$ be expressed in [[Definition:Decimal Notation|decimal notation]] as:
:$n = 10 x + y$
where:
: $x > y$
:$0 \le x \le 9, 0 \le y \le 9$
The [[Definition:Reversal|reversal]] of $n$ is $10 y + x$.
... | Kaprekar's Process for 2-Digit Numbers | https://proofwiki.org/wiki/Kaprekar's_Process_for_2-Digit_Numbers | https://proofwiki.org/wiki/Kaprekar's_Process_for_2-Digit_Numbers | [
"Kaprekar's Process",
"Reversals"
] | [
"Definition:Kaprekar's Process",
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Zero Digit"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Decimal Notation",
"Definition:Reversal",
"Definition:Kaprekar's Process",
"Definition:Reversal",
"Definition:Kaprekar's Process",
"Definition:Multiple/Integer"
] |
proofwiki-13078 | Necessary and Sufficient Condition for First Order System to be Field for Functional | Let $\mathbf y$ be an N-dimensional vector.
Let $J$ be a (real) functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let the corresponding momenta and Hamiltonian be:
{{begin-eqn}}
{{eqn | l = \map {\mathbf p} {x, \mathbf y, \mathbf y'}
| r = \dfrac {\partial \map F... | === Necessary Condition ===
Set $\mathbf y = \map \bspsi {x, \mathbf y}$ in the definition of momenta and Hamiltonian.
Substitute corresponding definitions into the consistency relation.
On the {{LHS}} we have:
:$\dfrac {\partial \map {p_i} {x, \mathbf y, \map \bspsi {x, \mathbf y} } } {\partial x} = \dfrac {\partial^2... | Let $\mathbf y$ be an [[Definition:Dimension|N-dimensional]] [[Definition:Vector|vector]].
Let $J$ be a [[Definition:Real Functional|(real) functional]] such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let the corresponding [[Definition:Canonical Variable|momenta]] and [[Def... | === Necessary Condition ===
Set $\mathbf y = \map \bspsi {x, \mathbf y}$ in the definition of [[Definition:Canonical Variable|momenta]] and [[Definition:Hamiltonian|Hamiltonian]].
Substitute corresponding definitions into the [[Euler's Equation for Vanishing Variation in Canonical Variables|consistency]] relation.
O... | Necessary and Sufficient Condition for First Order System to be Field for Functional | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Field_for_Functional | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Field_for_Functional | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Functional/Real",
"Definition:Canonical Variable",
"Definition:Hamiltonian",
"Definition:Boundary Condition",
"Definition:Boundary Condition",
"Definition:Functional/Real",
"Definition:Self-Adjoint Boundary Conditions",
"Euler's Equation for... | [
"Definition:Canonical Variable",
"Definition:Hamiltonian",
"Euler's Equation for Vanishing Variation in Canonical Variables",
"Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint",
"Definition:Depend",
"Definition:Vector",
"Definition:Variable",
"Definition:Depend",
"Defini... |
proofwiki-13079 | Numbers with 6 or more Prime Factors | The sequence of positive integers with $6$ or more prime factors (not necessarily distinct) begins:
:$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \ldots$
{{OEIS|A046305}} | {{begin-eqn}}
{{eqn | l = 64
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2
| c =
}}
{{eqn | l = 96
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 3
| c =
}}
{{eqn | l = 128
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2}
| c =
}}
{{eqn | l = 144
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] with $6$ or more [[Definition:Prime Factor|prime factors]] (not necessarily [[Definition:Distinct|distinct]]) begins:
:$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \ldots$
{{OEIS|A046305}} | {{begin-eqn}}
{{eqn | l = 64
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2
| c =
}}
{{eqn | l = 96
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 3
| c =
}}
{{eqn | l = 128
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times 2}
| c =
}}
{{eqn | l = 144
... | Numbers with 6 or more Prime Factors | https://proofwiki.org/wiki/Numbers_with_6_or_more_Prime_Factors | https://proofwiki.org/wiki/Numbers_with_6_or_more_Prime_Factors | [
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Prime Factor",
"Definition:Distinct"
] | [] |
proofwiki-13080 | Cube as Sum of Sequence of Centered Hexagonal Numbers | :$C_n = \ds \sum_{i \mathop = 1}^n H_i$
where:
:$C_n$ denotes the $n$th cube number
:$H_i$ denotes the $i$th centered hexagonal number. | From Closed Form for Centered Hexagonal Numbers:
:$H_n = 3 n \paren {n - 1} + 1$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n H_i
| r = \sum_{i \mathop = 1}^n \paren {3 i \paren {i - 1} + 1}
| c =
}}
{{eqn | r = \sum_{i \mathop = 1}^n \paren {3 i^2 - 3 i + 1}
| c =
}}
{{eqn | r = 3 \sum_{... | :$C_n = \ds \sum_{i \mathop = 1}^n H_i$
where:
:$C_n$ denotes the $n$th [[Definition:Cube Number|cube number]]
:$H_i$ denotes the $i$th [[Definition:Centered Hexagonal Number|centered hexagonal number]]. | From [[Closed Form for Centered Hexagonal Numbers]]:
:$H_n = 3 n \paren {n - 1} + 1$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n H_i
| r = \sum_{i \mathop = 1}^n \paren {3 i \paren {i - 1} + 1}
| c =
}}
{{eqn | r = \sum_{i \mathop = 1}^n \paren {3 i^2 - 3 i + 1}
| c =
}}
{{eqn | r = 3 ... | Cube as Sum of Sequence of Centered Hexagonal Numbers | https://proofwiki.org/wiki/Cube_as_Sum_of_Sequence_of_Centered_Hexagonal_Numbers | https://proofwiki.org/wiki/Cube_as_Sum_of_Sequence_of_Centered_Hexagonal_Numbers | [
"Centered Hexagonal Numbers",
"Cube Numbers",
"Cube as Sum of Sequence of Centered Hexagonal Numbers"
] | [
"Definition:Cube Number",
"Definition:Centered Hexagonal Number"
] | [
"Closed Form for Centered Hexagonal Numbers",
"Sum of Sequence of Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-13081 | Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared | Let $p$ be a prime number.
Then there exists at least one positive integer $n$ greater than $1$ such that:
:$n^{p - 1} \equiv 1 \pmod {p^2}$ | {{begin-eqn}}
{{eqn | l = p^2
| o = \equiv
| r = 0
| rr = \pmod {p^2}
}}
{{eqn | l = 1
| o = \equiv
| r = 1
| rr = \pmod {p^2}
}}
{{eqn | ll= \leadsto
| l = p^2 + 1
| o = \equiv
| r = 0 + 1
| rr= \pmod {p^2}
| c = Modulo Addition is Well-Defined
}}
{{eqn... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then there exists at least one [[Definition:Positive Integer|positive integer]] $n$ greater than $1$ such that:
:$n^{p - 1} \equiv 1 \pmod {p^2}$ | {{begin-eqn}}
{{eqn | l = p^2
| o = \equiv
| r = 0
| rr = \pmod {p^2}
}}
{{eqn | l = 1
| o = \equiv
| r = 1
| rr = \pmod {p^2}
}}
{{eqn | ll= \leadsto
| l = p^2 + 1
| o = \equiv
| r = 0 + 1
| rr= \pmod {p^2}
| c = [[Modulo Addition is Well-Defined]]
}}
{... | Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared | https://proofwiki.org/wiki/Existence_of_Number_to_Power_of_Prime_Minus_1_less_1_divisible_by_Prime_Squared | https://proofwiki.org/wiki/Existence_of_Number_to_Power_of_Prime_Minus_1_less_1_divisible_by_Prime_Squared | [
"Number Theory",
"Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared"
] | [
"Definition:Prime Number",
"Definition:Positive/Integer"
] | [
"Modulo Addition is Well-Defined",
"Congruence of Powers"
] |
proofwiki-13082 | Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes | The smallest positive integer which is both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers is $65$:
{{begin-eqn}}
{{eqn | l = 65
| m = 16 + 49
| mo= =
| r = 4^2 + 7^2
| c =
}}
{{eqn | m = 1 + 64
| mo= =
| r = 1^2 + 8^2
| c =
}}
{{eqn |... | From Sum of 2 Squares in 2 Distinct Ways, the smallest $2$ positive integer which are the sum of $2$ square numbers in two distinct ways are $50$ and $65$.
But $50$ cannot be expressed as the sum of $2$ cube numbers:
{{begin-eqn}}
{{eqn | l = 50 - 1^3
| r = 49
| c = which is not cubic
}}
{{eqn | l = 50 - 2^... | The smallest [[Definition:Positive Integer|positive integer]] which is both the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways and also the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cube numbers]] is $65$:
{{... | From [[Sum of 2 Squares in 2 Distinct Ways]], the smallest $2$ [[Definition:Positive Integer|positive integer]] which are the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways are $50$ and $65$.
But $50$ cannot be expressed as the [[Defi... | Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways_which_is_also_Sum_of_Cubes | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways_which_is_also_Sum_of_Cubes | [
"Sums of Squares",
"Sums of Cubes",
"65",
"Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [
"Sum of 2 Squares in 2 Distinct Ways",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct",
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Cube Number",
"Definition:Cube Number",
"Definition:Cube Number"
] |
proofwiki-13083 | Magic Constant of Order 5 Magic Square | The magic constant of the order $5$ magic square is $65$. | Let $M_5$ denote an order $5$ magic square
By Sum of Terms of Magic Square, the total of all the entries in $M_5$ is given by:
:$T_5 = \dfrac {5^2 \paren {5^2 + 1} } 2 = \dfrac {25 \times 26} 2 = 325$
As there are $5$ rows of $M_5$, the magic constant of $M_5$ is given by:
:$S_5 = \dfrac {325} 5 = 65$
{{qed}} | The [[Definition:Magic Constant|magic constant]] of the [[Magic Square/Examples/Order 5|order $5$ magic square]] is $65$. | Let $M_5$ denote an [[Magic Square/Examples/Order 5|order $5$ magic square]]
By [[Sum of Terms of Magic Square]], the total of all the entries in $M_5$ is given by:
:$T_5 = \dfrac {5^2 \paren {5^2 + 1} } 2 = \dfrac {25 \times 26} 2 = 325$
As there are $5$ [[Definition:Row of Matrix|rows]] of $M_5$, the [[Definition:M... | Magic Constant of Order 5 Magic Square/Proof 1 | https://proofwiki.org/wiki/Magic_Constant_of_Order_5_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Order_5_Magic_Square/Proof_1 | [
"Magic Squares",
"65",
"Magic Constant of Order 5 Magic Square"
] | [
"Definition:Magic Square/Magic Constant",
"Magic Square/Examples/Order 5"
] | [
"Magic Square/Examples/Order 5",
"Sum of Terms of Magic Square",
"Definition:Matrix/Row",
"Definition:Magic Square/Magic Constant"
] |
proofwiki-13084 | Triple of Triangular Numbers whose Pairwise Sums are Triangular | The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers:
:$66, 105, 105$ | Throughout we use Closed Form for Triangular Numbers, which gives that the $n$th triangular number can be expressed as:
:$T_n = \dfrac {11 \times 12} 2$
We have:
{{begin-eqn}}
{{eqn | l = 66
| r = \frac {11 \times 12} 2
| c = and so is triangular
}}
{{eqn | l = 105
| r = \frac {14 \times 15} 2
|... | The following [[Definition:Ordered Triple|triplet]] of [[Definition:Triangular Number|triangular numbers]] has the property that the [[Definition:Integer Addition|sum]] of each [[Definition:Ordered Pair|pair]] of them, and their total, are all [[Definition:Triangular Number|triangular numbers]]:
:$66, 105, 105$ | Throughout we use [[Closed Form for Triangular Numbers]], which gives that the $n$th [[Definition:Triangular Number|triangular number]] can be expressed as:
:$T_n = \dfrac {11 \times 12} 2$
We have:
{{begin-eqn}}
{{eqn | l = 66
| r = \frac {11 \times 12} 2
| c = and so is [[Definition:Triangular Number|tr... | Triple of Triangular Numbers whose Pairwise Sums are Triangular | https://proofwiki.org/wiki/Triple_of_Triangular_Numbers_whose_Pairwise_Sums_are_Triangular | https://proofwiki.org/wiki/Triple_of_Triangular_Numbers_whose_Pairwise_Sums_are_Triangular | [
"Triangular Numbers"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Triangular Number",
"Definition:Addition/Integers",
"Definition:Ordered Pair",
"Definition:Triangular Number"
] | [
"Closed Form for Triangular Numbers",
"Definition:Triangular Number",
"Definition:Triangular Number",
"Definition:Triangular Number",
"Definition:Triangular Number",
"Definition:Triangular Number",
"Definition:Triangular Number"
] |
proofwiki-13085 | Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways | Let $n \in \Z_{>0}$ be a positive even integer.
Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways.
Then $n$ belongs to the set:
:$\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$
{{OEI... | The sequence of odd positive composite integers begins:
:$9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, \ldots$
which is more than we need for this proof.
{{OEIS|A071904}}
Generating the possible sums of these proceeds as follows:
{{begin-eqn}}
{{eqn | l = 18
| r = 9 + 9
| c =
}}
{{eqn | l = 2... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive]] [[Definition:Even Integer|even integer]].
Let $n$ be such that it cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integ... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composite Number|composite integers]] begins:
:$9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, \ldots$
which is more than we need for this proof.
{{OEIS|A071904}}
Generatin... | Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways | https://proofwiki.org/wiki/Positive_Even_Integers_as_Sum_of_2_Composite_Odd_Integers_in_2_Ways | https://proofwiki.org/wiki/Positive_Even_Integers_as_Sum_of_2_Composite_Odd_Integers_in_2_Ways | [
"Composite Numbers",
"Odd Integers"
] | [
"Definition:Positive/Integer",
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Set"
] | [
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Positive/Intege... |
proofwiki-13086 | Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers | Let $k \in \Z_{>0}$ be a (strictly) positive integer.
The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$. | Let $n$ be an even integer greater than $18 k + 20$.
Then $n - 9 \paren {2 k - 2}$ is an even integer greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.
By Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers, every even integer greater than $38$ can be expressed as the sum of $2$ odd positive comp... | Let $k \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
The largest [[Definition:Even Integer|even integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of $2 k$ [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive]] [[Definition:Composit... | Let $n$ be an [[Definition:Even Integer|even integer]] greater than $18 k + 20$.
Then $n - 9 \paren {2 k - 2}$ is an [[Definition:Even Integer|even integer]] greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.
By [[Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers]], every [[Definition:Even Int... | Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers | https://proofwiki.org/wiki/Largest_Even_Integer_not_expressible_as_Sum_of_2_k_Odd_Composite_Integers | https://proofwiki.org/wiki/Largest_Even_Integer_not_expressible_as_Sum_of_2_k_Odd_Composite_Integers | [
"Composite Numbers",
"Odd Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Composite Number"
] | [
"Definition:Even Integer",
"Definition:Even Integer",
"Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers",
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Addition/In... |
proofwiki-13087 | Number whose Square and Cube use all Digits Once | The only integer whose square and cube use each of the digits from $0$ to $9$ exactly once each is $69$. | First the limits are established for the square and cube of an integer to have exactly $10$ digits between them.
{{begin-eqn}}
{{eqn | l = 46^2
| r = 2116
| c =
}}
{{eqn | l = 46^3
| r = 97 \, 336
| c =
}}
{{eqn | l = 47^2
| r = 2209
| c =
}}
{{eqn | l = 47^3
| r = 103 \, 82... | The only [[Definition:Integer|integer]] whose [[Definition:Square (Algebra)|square]] and [[Definition:Cube (Algebra)|cube]] use each of the [[Definition:Digit|digits]] from $0$ to $9$ exactly once each is $69$. | First the limits are established for the [[Definition:Square (Algebra)|square]] and [[Definition:Cube (Algebra)|cube]] of an [[Definition:Integer|integer]] to have exactly $10$ [[Definition:Digit|digits]] between them.
{{begin-eqn}}
{{eqn | l = 46^2
| r = 2116
| c =
}}
{{eqn | l = 46^3
| r = 97 \, 3... | Number whose Square and Cube use all Digits Once | https://proofwiki.org/wiki/Number_whose_Square_and_Cube_use_all_Digits_Once | https://proofwiki.org/wiki/Number_whose_Square_and_Cube_use_all_Digits_Once | [
"Recreational Mathematics",
"Square Numbers",
"Cube Numbers",
"69"
] | [
"Definition:Integer",
"Definition:Square/Function",
"Definition:Cube/Algebra",
"Definition:Digit"
] | [
"Definition:Square/Function",
"Definition:Cube/Algebra",
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Square/Function",
"Definition:Cube/Algebra",
"Definition:Digit",
"Definition:Digit",
"Definitio... |
proofwiki-13088 | Cube of 71 is Odd Integers in Sequence | The cube of $71$, when expressed in decimal notation, is the odd integers from $3$ to $11$ written in sequence:
::$71^3 = 357 \, 911$ | Verified by direct calculation. | The [[Definition:Cube (Algebra)|cube]] of $71$, when expressed in [[Definition:Decimal Notation|decimal notation]], is the [[Definition:Odd Integer|odd integers]] from $3$ to $11$ written in sequence:
::$71^3 = 357 \, 911$ | Verified by direct calculation. | Cube of 71 is Odd Integers in Sequence | https://proofwiki.org/wiki/Cube_of_71_is_Odd_Integers_in_Sequence | https://proofwiki.org/wiki/Cube_of_71_is_Odd_Integers_in_Sequence | [
"Cube Numbers",
"Odd Integers",
"71"
] | [
"Definition:Cube/Algebra",
"Definition:Decimal Notation",
"Definition:Odd Integer"
] | [] |
proofwiki-13089 | Prime Numbers which Divide Sum of All Lesser Primes | The following sequence of prime numbers has the property that each is a divisor of the sum of all primes equal to or smaller than them:
:$2$, $5$, $71$, $369 \, 119$, $415 \, 074 \, 643$, $55 \, 691 \, 042 \, 365 \, 834 \, 801$
{{OEIS|A007506}}
As of the time of the last edit to this page (April $2025$), no others are ... | Verified by calculation. | The following [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] has the property that each is a [[Definition:Divisor of Integer|divisor]] of the [[Definition:Integer Addition|sum]] of all [[Definition:Prime Number|primes]] equal to or smaller than them:
:$2$, $5$, $71$, $369 \, 119$,... | Verified by calculation. | Prime Numbers which Divide Sum of All Lesser Primes | https://proofwiki.org/wiki/Prime_Numbers_which_Divide_Sum_of_All_Lesser_Primes | https://proofwiki.org/wiki/Prime_Numbers_which_Divide_Sum_of_All_Lesser_Primes | [
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Prime Number"
] | [] |
proofwiki-13090 | 4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value | The following sets of $4$ positive integers which form an arithmetic sequence are the smallest which all have the same Euler $\phi$ value:
:$72, 78, 84, 90$
:$216, 222, 228, 234$
:$76 \, 326, 76 \, 332, 76 \, 338, 76 \, 344$ | {{begin-eqn}}
{{eqn | l = 78 - 72
| r = 6
| c =
}}
{{eqn | l = 84 - 78
| r = 6
| c =
}}
{{eqn | l = 90 - 84
| r = 6
| c =
}}
{{end-eqn}}
demonstrating that this is indeed an arithmetic sequence, with a common difference of $6$.
Now we show:
{{begin-eqn}}
{{eqn | l = \map \phi {72}... | The following [[Definition:Set|sets]] of $4$ [[Definition:Positive Integer|positive integers]] which form an [[Definition:Arithmetic Sequence|arithmetic sequence]] are the smallest which all have the same [[Definition:Euler Phi Function|Euler $\phi$ value]]:
:$72, 78, 84, 90$
:$216, 222, 228, 234$
:$76 \, 326, 76 \, 33... | {{begin-eqn}}
{{eqn | l = 78 - 72
| r = 6
| c =
}}
{{eqn | l = 84 - 78
| r = 6
| c =
}}
{{eqn | l = 90 - 84
| r = 6
| c =
}}
{{end-eqn}}
demonstrating that this is indeed an [[Definition:Arithmetic Sequence|arithmetic sequence]], with a [[Definition:Common Difference|common diffe... | 4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value | https://proofwiki.org/wiki/4_Positive_Integers_in_Arithmetic_Sequence_which_have_Same_Euler_Phi_Value | https://proofwiki.org/wiki/4_Positive_Integers_in_Arithmetic_Sequence_which_have_Same_Euler_Phi_Value | [
"Euler Phi Function"
] | [
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Arithmetic Sequence",
"Definition:Euler Phi Function"
] | [
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference"
] |
proofwiki-13091 | Positive Integers which are Euler Phi Value for 17 Integers | There are $17$ positive integers which have an Euler $\phi$ value of the following:
:$72, 96, 120, \ldots$ | From Numbers with Euler Phi Value of 72, it is seen that $17$ positive integers have an Euler $\phi$ value of $72$:
:$73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$
{{proof wanted|boring}} | There are $17$ [[Definition:Positive Integer|positive integers]] which have an [[Definition:Euler Phi Function|Euler $\phi$ value]] of the following:
:$72, 96, 120, \ldots$ | From [[Numbers with Euler Phi Value of 72]], it is seen that $17$ [[Definition:Positive Integer|positive integers]] have an [[Definition:Euler Phi Function|Euler $\phi$ value]] of $72$:
:$73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$
{{proof wanted|boring}} | Positive Integers which are Euler Phi Value for 17 Integers | https://proofwiki.org/wiki/Positive_Integers_which_are_Euler_Phi_Value_for_17_Integers | https://proofwiki.org/wiki/Positive_Integers_which_are_Euler_Phi_Value_for_17_Integers | [
"Euler Phi Function"
] | [
"Definition:Positive/Integer",
"Definition:Euler Phi Function"
] | [
"Numbers with Euler Phi Value of 72",
"Definition:Positive/Integer",
"Definition:Euler Phi Function"
] |
proofwiki-13092 | Euler Phi Function of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Let $\map \phi n$ denote the Euler $\phi$ function.
Then:
:$\map \phi n = \paren {p - 1} \paren {q - 1}$ | As $p$ and $q$ are distinct prime numbers, it follows that $p$ and $q$ are coprime.
Thus by Euler Phi Function is Multiplicative:
:$\map \phi n = \map \phi p \, \map \phi q$
From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
:$\map \phi q = q - 1$
Hence the result.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
Then:
:$\map \phi n = \paren {p - 1} \paren {q - 1}$ | As $p$ and $q$ are [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]], it follows that $p$ and $q$ are [[Definition:Coprime Integers|coprime]].
Thus by [[Euler Phi Function is Multiplicative]]:
:$\map \phi n = \map \phi p \, \map \phi q$
From [[Euler Phi Function of Prime]]:
:$\map \phi p = p ... | Euler Phi Function of Non-Square Semiprime/Proof 1 | https://proofwiki.org/wiki/Euler_Phi_Function_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Euler_Phi_Function_of_Non-Square_Semiprime/Proof_1 | [
"Euler Phi Function",
"Semiprimes",
"Euler Phi Function of Non-Square Semiprime"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Euler Phi Function"
] | [
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Euler Phi Function is Multiplicative",
"Euler Phi Function of Prime"
] |
proofwiki-13093 | Euler Phi Function of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Let $\map \phi n$ denote the Euler $\phi$ function.
Then:
:$\map \phi n = \paren {p - 1} \paren {q - 1}$ | A semiprime with distinct prime factors is a square-free integer.
:$\map \phi n = \ds \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
As there are $2$ prime factors: $p$ and $q$, this devolves to:
:$\map \phi n = \paren {p - 1} \paren ... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
Then:
:$\map \phi n = \paren {p - 1} \paren {q - 1}$ | A [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] is a [[Definition:Square-Free|square-free]] [[Definition:Integer|integer]].
:$\map \phi n = \ds \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ den... | Euler Phi Function of Non-Square Semiprime/Proof 2 | https://proofwiki.org/wiki/Euler_Phi_Function_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Euler_Phi_Function_of_Non-Square_Semiprime/Proof_2 | [
"Euler Phi Function",
"Semiprimes",
"Euler Phi Function of Non-Square Semiprime"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Euler Phi Function"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Square-Free",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Factor"
] |
proofwiki-13094 | Euler Phi Function of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let $\map \phi n$ be the Euler $\phi$ function of $n$.
That is, let $\map \phi n$ be the count of strictly positive integers less than or equal to $n$ which are prime to $n$.
Then:
:$\map \phi n = \ds \prod_{\substack {p \mathop \divides n \\ p \mathop ... | We have that the Euler Phi Function is Multiplicative.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
From the definition of prime number, each of the prime factors of $n$ is coprime to all other divisors of $n$.
From Euler Phi Function of Prime, we hav... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let $\map \phi n$ be the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $n$.
That is, let $\map \phi n$ be the count of [[Definition:Strictly Positive|strictly positive integers]] less tha... | We have that the [[Euler Phi Function is Multiplicative]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
From the definition of [[Definition:Prime Number|prime number]], each of the [[Definition:Prime Factor|prime f... | Euler Phi Function of Square-Free Integer/Proof 1 | https://proofwiki.org/wiki/Euler_Phi_Function_of_Square-Free_Integer | https://proofwiki.org/wiki/Euler_Phi_Function_of_Square-Free_Integer/Proof_1 | [
"Euler Phi Function",
"Euler Phi Function of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Euler Phi Function",
"Definition:Strictly Positive",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Euler Phi Function is Multiplicative",
"Definition:Prime Decomposition",
"Definition:Prime Number",
"Definition:Prime Factor",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer",
"Euler Phi Function of Prime",
"Definition:Prime Number"
] |
proofwiki-13095 | Euler Phi Function of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let $\map \phi n$ be the Euler $\phi$ function of $n$.
That is, let $\map \phi n$ be the count of strictly positive integers less than or equal to $n$ which are prime to $n$.
Then:
:$\map \phi n = \ds \prod_{\substack {p \mathop \divides n \\ p \mathop ... | From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
As $n$ is square-free:
:$\ds n = \prod_{p \mathop \divides n} p$
Hence:
:$\ds \map \phi n = \prod_{p \mathop \divides n} p \paren {1 - \frac 1 p}$
and so:
:$\ds \map \phi n = \prod_{p \mathop \divides n} \pare... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let $\map \phi n$ be the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $n$.
That is, let $\map \phi n$ be the count of [[Definition:Strictly Positive|strictly positive integers]] less tha... | From [[Euler Phi Function of Integer]]:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
As $n$ is [[Definition:Square-Free Integer|square-free]]:
:$\ds n = \prod_{p \mathop \divides n} p$
Hence:
:$\ds \map \phi n = \prod_{p \mathop \divides n} p \paren {1 - \frac 1 p}$
and so:
:$\ds \ma... | Euler Phi Function of Square-Free Integer/Proof 2 | https://proofwiki.org/wiki/Euler_Phi_Function_of_Square-Free_Integer | https://proofwiki.org/wiki/Euler_Phi_Function_of_Square-Free_Integer/Proof_2 | [
"Euler Phi Function",
"Euler Phi Function of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Euler Phi Function",
"Definition:Strictly Positive",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Euler Phi Function of Integer",
"Definition:Square-Free Integer"
] |
proofwiki-13096 | Euler Phi Function of 2 times Odd Prime | Let $n \in \Z_{>0}$ be a semiprime of the form $2 p$, where $p$ is an odd prime.
Let $\map \phi n$ denote the Euler $\phi$ function.
Then:
:$\map \phi n = p - 1$ | By definition $n$ is a semiprime.
As $p$ is an odd prime, $n$ is not square.
Thus from Euler Phi Function of Non-Square Semiprime:
:$\map \phi n = \paren {2 - 1} \paren {p - 1}$
Hence the result.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] of the form $2 p$, where $p$ is an [[Definition:Odd Prime|odd prime]].
Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
Then:
:$\map \phi n = p - 1$ | By definition $n$ is a [[Definition:Semiprime Number|semiprime]].
As $p$ is an [[Definition:Odd Prime|odd prime]], $n$ is not [[Definition:Square Number|square]].
Thus from [[Euler Phi Function of Non-Square Semiprime]]:
:$\map \phi n = \paren {2 - 1} \paren {p - 1}$
Hence the result.
{{qed}} | Euler Phi Function of 2 times Odd Prime | https://proofwiki.org/wiki/Euler_Phi_Function_of_2_times_Odd_Prime | https://proofwiki.org/wiki/Euler_Phi_Function_of_2_times_Odd_Prime | [
"Euler Phi Function",
"Semiprimes",
"Euler Phi Function of 2 times Odd Prime"
] | [
"Definition:Semiprime Number",
"Definition:Odd Prime",
"Definition:Euler Phi Function"
] | [
"Definition:Semiprime Number",
"Definition:Odd Prime",
"Definition:Square Number",
"Euler Phi Function of Non-Square Semiprime"
] |
proofwiki-13097 | Set is Not Element of Itself | There cannot exist a set which is an element of itself.
That is:
:$\neg \exists a: a \in a$ | {{AimForCont}} $a$ is such a set.
Then $a \in a$ and $a \in \set a$.
$a \ne \O$ because the empty set has no elements by definition.
It is also seen that:
{{begin-eqn}}
{{eqn | l = a \cap \set a
| r = \set {x: x \in a \land x \in \set a}
| c = {{Defof|Set Intersection}}
}}
{{eqn | r = \set {x: x \in a \land... | There cannot exist a [[Definition:Set|set]] which is an [[Definition:Element|element]] of itself.
That is:
:$\neg \exists a: a \in a$ | {{AimForCont}} $a$ is such a set.
Then $a \in a$ and $a \in \set a$.
$a \ne \O$ because the [[Definition:Empty Set|empty set]] has no [[Definition:Element|elements]] by definition.
It is also seen that:
{{begin-eqn}}
{{eqn | l = a \cap \set a
| r = \set {x: x \in a \land x \in \set a}
| c = {{Defof|Set ... | Set is Not Element of Itself | https://proofwiki.org/wiki/Set_is_Not_Element_of_Itself | https://proofwiki.org/wiki/Set_is_Not_Element_of_Itself | [
"Axiom of Foundation"
] | [
"Definition:Set",
"Definition:Element"
] | [
"Definition:Empty Set",
"Definition:Element",
"Axiom:Axiom of Foundation",
"Definition:Contradiction",
"Definition:Set",
"Category:Axiom of Foundation"
] |
proofwiki-13098 | Product of Number of Edges, Edges per Face and Faces of Tetrahedron | The product of the number of edges, edges per face and faces of a tetrahedron is $72$. | A tetrahedron has:
:$6$ edges
and:
:$4$ faces.
Each face is a triangle, and so has $3$ edges.
Hence:
:$6 \times 4 \times 3 = 72$
{{qed}} | The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Tetrahedron|tetrahedron]] is $72$. | A [[Definition:Tetrahedron|tetrahedron]] has:
:$6$ [[Definition:Edge of Polyhedron|edges]]
and:
:$4$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle (Geometry)|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
Hence:
:$6 \times 4 \times... | Product of Number of Edges, Edges per Face and Faces of Tetrahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Tetrahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Tetrahedron | [
"Tetrahedra",
"72"
] | [
"Definition:Multiplication/Integers",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Tetrahedron"
] | [
"Definition:Tetrahedron",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Triangle (Geometry)",
"Definition:Polyhedron/Edge"
] |
proofwiki-13099 | Product of Number of Edges, Edges per Face and Faces of Cube | The product of the number of edges, edges per face and faces of a cube is $288$. | A cube has:
:$12$ edges
and:
:$6$ faces.
Each face is a square, and so has $4$ edges.
Hence:
:$12 \times 6 \times 4 = 288$
{{qed}} | The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Cube (Geometry)|cube]] is $288$. | A [[Definition:Cube (Geometry)|cube]] has:
:$12$ [[Definition:Edge of Polyhedron|edges]]
and:
:$6$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Square (Geometry)|square]], and so has $4$ [[Definition:Edge of Polyhedron|edges]].
Hence:
:$12 \times 6 \times 4 = ... | Product of Number of Edges, Edges per Face and Faces of Cube | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Cube | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Cube | [
"Cubes",
"288"
] | [
"Definition:Multiplication/Integers",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Cube/Geometry"
] | [
"Definition:Cube/Geometry",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Quadrilateral/Square",
"Definition:Polyhedron/Edge"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.