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-12200 | Euler Phi Function of n equal to Euler Phi Function of n+3 | Let $\phi$ denote the Euler $\phi$ function.
The only solutions to the equation:
:$\map \phi n = \map \phi {n + 3}$
less than $1 \, 000 \, 000$ are:
:$\map \phi 3 = \map \phi 6 = 2$
:$\map \phi 5 = \map \phi 8 = 4$ | {{ProofWanted|By exhaustion, I suppose}} | Let $\phi$ denote the [[Definition:Euler Phi Function|Euler $\phi$ function]].
The only solutions to the equation:
:$\map \phi n = \map \phi {n + 3}$
less than $1 \, 000 \, 000$ are:
:$\map \phi 3 = \map \phi 6 = 2$
:$\map \phi 5 = \map \phi 8 = 4$ | {{ProofWanted|By exhaustion, I suppose}} | Euler Phi Function of n equal to Euler Phi Function of n+3 | https://proofwiki.org/wiki/Euler_Phi_Function_of_n_equal_to_Euler_Phi_Function_of_n+3 | https://proofwiki.org/wiki/Euler_Phi_Function_of_n_equal_to_Euler_Phi_Function_of_n+3 | [
"Euler Phi Function"
] | [
"Definition:Euler Phi Function"
] | [] |
proofwiki-12201 | Friendship Theorem | Let there be a group of $6$ people.
The traditional setting is that these $6$ people are at a party.
Then (at least) one of the following $2$ statements is true:
:$(1): \quad$ At least $3$ of these $6$ people have all met each other before
:$(2): \quad$ At least $3$ of these $6$ people have never met each other before.... | This is a simple example of Ramsey's Theorem.
{{ProofWanted}} | Let there be a group of $6$ people.
The traditional setting is that these $6$ people are at a party.
Then (at least) one of the following $2$ [[Definition:Statement|statements]] is [[Definition:True|true]]:
:$(1): \quad$ At least $3$ of these $6$ people have all met each other before
:$(2): \quad$ At least $3$ of ... | This is a simple example of [[Ramsey's Theorem]].
{{ProofWanted}} | Friendship Theorem/Proof 1 | https://proofwiki.org/wiki/Friendship_Theorem | https://proofwiki.org/wiki/Friendship_Theorem/Proof_1 | [
"Ramsey Theory",
"Classic Problems",
"Graph Colorings",
"Friendship Theorem",
"Named Theorems"
] | [
"Definition:Statement",
"Definition:True"
] | [
"Ramsey's Theorem"
] |
proofwiki-12202 | Volume of Smallest Tetrahedron with Integer Edges and Integer Volume | The volume of the smallest tetrahedron with integer edges and integer volume is $3$.
There are $2$ possible sets of edges:
:$32, 33, 35, 44, 70, 76$
:$21, 32, 47, 56, 58, 76$ | {{tidy}}
Tartaglia's Formula gives us that:
:$V_T^2 = \dfrac {1} {288} \det \ \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$
where:
:$V_T$ denotes th... | The [[Definition:Volume|volume]] of the smallest [[Definition:Tetrahedron|tetrahedron]] with [[Definition:Integer|integer]] [[Definition:Edge of Polyhedron|edges]] and [[Definition:Integer|integer]] [[Definition:Volume|volume]] is $3$.
There are $2$ possible [[Definition:Set|sets]] of [[Definition:Edge of Polyhedron|e... | {{tidy}}
[[Tartaglia's Formula]] gives us that:
:$V_T^2 = \dfrac {1} {288} \det \ \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$
where:
:$V_T$ deno... | Volume of Smallest Tetrahedron with Integer Edges and Integer Volume | https://proofwiki.org/wiki/Volume_of_Smallest_Tetrahedron_with_Integer_Edges_and_Integer_Volume | https://proofwiki.org/wiki/Volume_of_Smallest_Tetrahedron_with_Integer_Edges_and_Integer_Volume | [
"Tetrahedra"
] | [
"Definition:Volume",
"Definition:Tetrahedron",
"Definition:Integer",
"Definition:Polyhedron/Edge",
"Definition:Integer",
"Definition:Volume",
"Definition:Set",
"Definition:Polyhedron/Edge"
] | [
"Tartaglia's Formula",
"Definition:Volume",
"Definition:Tetrahedron"
] |
proofwiki-12203 | Ramanujan's Infinite Nested Roots | :$3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }$ | We have:
{{begin-eqn}}
{{eqn | l = 3
| r = \sqrt 9
| c =
}}
{{eqn | r = \sqrt {1 + 8}
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {16} }
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {1 + 15} }
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt {25} } }
| c =
}}
{{eqn | r = \sqrt {1 + 2... | :$3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }$ | We have:
{{begin-eqn}}
{{eqn | l = 3
| r = \sqrt 9
| c =
}}
{{eqn | r = \sqrt {1 + 8}
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {16} }
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {1 + 15} }
| c =
}}
{{eqn | r = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt {25} } }
| c =
}}
{{eqn | r = \sqrt {1 + ... | Ramanujan's Infinite Nested Roots | https://proofwiki.org/wiki/Ramanujan's_Infinite_Nested_Roots | https://proofwiki.org/wiki/Ramanujan's_Infinite_Nested_Roots | [
"Number Theory"
] | [] | [
"Square of Sum"
] |
proofwiki-12204 | Product of Three Consecutive Integers is never Perfect Power | Let $n \in \Z_{> 1}$ be a (strictly) positive integer.
Then:
:$\paren {n - 1} n \paren {n + 1}$
cannot be expressed in the form $a^k$ for $a, k \in \Z$ where $k \ge 2$.
That is, the product of $3$ consecutive (strictly) positive integers can never be a perfect power. | {{AimForCont}} $\paren {n - 1} n \paren {n + 1} = a^k$ for $a, k \in \Z$ where $k \ge 2$.
We have that:
:$\gcd \set {n − 1, n} = 1 = \gcd \set {n, n + 1}$
Thus $n$ must itself be a perfect power of the form $z^k$ for some $z \in \Z$.
That means $\paren {n - 1} \paren {n + 1} = n^2 - 1$ must also be a perfect power of t... | Let $n \in \Z_{> 1}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\paren {n - 1} n \paren {n + 1}$
cannot be expressed in the form $a^k$ for $a, k \in \Z$ where $k \ge 2$.
That is, the [[Definition:Multiplication|product]] of $3$ consecutive [[Definition:Strictly Positive Intege... | {{AimForCont}} $\paren {n - 1} n \paren {n + 1} = a^k$ for $a, k \in \Z$ where $k \ge 2$.
We have that:
:$\gcd \set {n − 1, n} = 1 = \gcd \set {n, n + 1}$
Thus $n$ must itself be a [[Definition:Perfect Power|perfect power]] of the form $z^k$ for some $z \in \Z$.
That means $\paren {n - 1} \paren {n + 1} = n^2 - 1$ m... | Product of Three Consecutive Integers is never Perfect Power | https://proofwiki.org/wiki/Product_of_Three_Consecutive_Integers_is_never_Perfect_Power | https://proofwiki.org/wiki/Product_of_Three_Consecutive_Integers_is_never_Perfect_Power | [
"Number Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Multiplication",
"Definition:Strictly Positive/Integer",
"Definition:Perfect Power"
] | [
"Definition:Perfect Power",
"Definition:Perfect Power",
"Definition:Integer",
"Proof by Contradiction"
] |
proofwiki-12205 | Barbier's Theorem | Let $K$ be a closed curve of constant diameter.
{{explain|"constant diameter"}}
Let the circumference of $K$ be $c$.
Let the diameter of $K$ be $d$.
Then:
:$\dfrac c d = \pi$ | {{ProofWanted}}
{{Namedfor|Joseph-Émile Barbier|cat = Barbier}} | Let $K$ be a [[Definition:Closed Curve|closed curve]] of constant [[Definition:Diameter of Geometric Figure|diameter]].
{{explain|"constant diameter"}}
Let the [[Definition:Circumference|circumference]] of $K$ be $c$.
Let the [[Definition:Diameter of Geometric Figure|diameter]] of $K$ be $d$.
Then:
:$\dfrac c d = \pi... | {{ProofWanted}}
{{Namedfor|Joseph-Émile Barbier|cat = Barbier}} | Barbier's Theorem | https://proofwiki.org/wiki/Barbier's_Theorem | https://proofwiki.org/wiki/Barbier's_Theorem | [
"Geometry"
] | [
"Definition:Closed Curve",
"Definition:Geometric Figure/Diameter",
"Definition:Circumference",
"Definition:Geometric Figure/Diameter"
] | [] |
proofwiki-12206 | Newton's Formula for Pi | $\pi$ (pi) can be approximated using the formula:
:$\pi = \dfrac {3 \sqrt 3} 4 + 24 \paren {\dfrac 2 {3 \times 2^3} - \dfrac 1 {5 \times 2^5} - \dfrac 1 {28 \times 2^7} - \dfrac 1 {72 \times 2^9} - \dfrac 5 {704 \times 2^{11} } - \dfrac 7 {1664 \times 2^{13} } - \cdots}$ | Let $\AA$ denote the area of the shaded region in the following diagram:
:500px
Consider the semicircle embedded in the cartesian plane:
:whose radius is $\dfrac 1 2$
and
:whose center is the point $\tuple {\dfrac 1 2, 0}$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {x - \frac 1 2}^2 + \paren {y - 0}^2
| r = \fra... | [[Definition:Pi|$\pi$ (pi)]] can be approximated using the formula:
:$\pi = \dfrac {3 \sqrt 3} 4 + 24 \paren {\dfrac 2 {3 \times 2^3} - \dfrac 1 {5 \times 2^5} - \dfrac 1 {28 \times 2^7} - \dfrac 1 {72 \times 2^9} - \dfrac 5 {704 \times 2^{11} } - \dfrac 7 {1664 \times 2^{13} } - \cdots}$ | Let $\AA$ denote the [[Definition:Area|area]] of the shaded [[Definition:Region of Plane|region]] in the following diagram:
:[[File:Newtons-Approximation-to-Pi.png|500px]]
Consider the [[Definition:Semicircle|semicircle]] embedded in the [[Definition:Cartesian Plane|cartesian plane]]:
:whose [[Definition:Radius of ... | Newton's Formula for Pi | https://proofwiki.org/wiki/Newton's_Formula_for_Pi | https://proofwiki.org/wiki/Newton's_Formula_for_Pi | [
"Formulas for Pi"
] | [
"Definition:Pi"
] | [
"Definition:Area",
"Definition:Region/Plane",
"File:Newtons-Approximation-to-Pi.png",
"Definition:Circle/Semicircle",
"Definition:Cartesian Plane",
"Definition:Circle/Radius",
"Definition:Circle/Semicircle/Center",
"Definition:Point",
"Equation of Circle",
"Binomial Theorem/General Binomial Theore... |
proofwiki-12207 | Machin's Formula for Pi | :$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | Let $\tan \alpha = \dfrac 1 5$.
Then:
{{begin-eqn}}
{{eqn | l = \tan 2 \alpha
| r = \frac {2 \tan \alpha} {1 - \tan^2 \alpha}
| c = Double Angle Formula for Tangent
}}
{{eqn | r = \frac {2 / 5} {1 - 1 / 25}
| c =
}}
{{eqn | r = \frac 5 {12}
| c =
}}
{{eqn | ll= \leadsto
| l = \tan 4 \alp... | :$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | Let $\tan \alpha = \dfrac 1 5$.
Then:
{{begin-eqn}}
{{eqn | l = \tan 2 \alpha
| r = \frac {2 \tan \alpha} {1 - \tan^2 \alpha}
| c = [[Double Angle Formula for Tangent]]
}}
{{eqn | r = \frac {2 / 5} {1 - 1 / 25}
| c =
}}
{{eqn | r = \frac 5 {12}
| c =
}}
{{eqn | ll= \leadsto
| l = \tan 4... | Machin's Formula for Pi/Proof 1 | https://proofwiki.org/wiki/Machin's_Formula_for_Pi | https://proofwiki.org/wiki/Machin's_Formula_for_Pi/Proof_1 | [
"Formulas for Pi",
"Machin's Formula for Pi"
] | [] | [
"Double Angle Formulas/Tangent",
"Double Angle Formulas/Tangent",
"Tangent of Difference",
"Tangent of 45 Degrees"
] |
proofwiki-12208 | Machin's Formula for Pi | :$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{begin-eqn}}
{{eqn | l = \map \arg {\paren {5 + i }^4 \paren {239 - i} }
| r = \map \arg {5 + i}^4 + \map \arg {239 - i}
| c = Argument of Product equals Sum of Arguments
}}
{{eqn | r = 4 \map \arg {5 + i} + \map \arg {239 - i}
| c = Argument of Product equals Sum of Arguments
}}
{{eqn | r = 4 \arcta... | :$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{begin-eqn}}
{{eqn | l = \map \arg {\paren {5 + i }^4 \paren {239 - i} }
| r = \map \arg {5 + i}^4 + \map \arg {239 - i}
| c = [[Argument of Product equals Sum of Arguments]]
}}
{{eqn | r = 4 \map \arg {5 + i} + \map \arg {239 - i}
| c = [[Argument of Product equals Sum of Arguments]]
}}
{{eqn | r = ... | Machin's Formula for Pi/Proof 2 | https://proofwiki.org/wiki/Machin's_Formula_for_Pi | https://proofwiki.org/wiki/Machin's_Formula_for_Pi/Proof_2 | [
"Formulas for Pi",
"Machin's Formula for Pi"
] | [] | [
"Argument of Product equals Sum of Arguments",
"Argument of Product equals Sum of Arguments",
"Inverse Tangent is Odd Function",
"Binomial Theorem"
] |
proofwiki-12209 | Ordinals under Addition form Ordered Semigroup | $\struct {\On, +, \le}$ forms an ordered semigroup, where:
:$\On$ denotes the class of all ordinals
:$+$ denotes ordinal addition. | The result follows from:
:Ordinals under Addition form Semigroup
:Subset is Compatible with Ordinal Addition.
{{qed}}
Category:Ordinal Arithmetic
8qttvuwa309nvykcf83jkgoc1e9zw3s | $\struct {\On, +, \le}$ forms an [[Definition:Ordered Semigroup|ordered semigroup]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$+$ denotes [[Definition:Ordinal Addition|ordinal addition]]. | The result follows from:
:[[Ordinals under Addition form Semigroup]]
:[[Subset is Compatible with Ordinal Addition]].
{{qed}}
[[Category:Ordinal Arithmetic]]
8qttvuwa309nvykcf83jkgoc1e9zw3s | Ordinals under Addition form Ordered Semigroup | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Ordered_Semigroup | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Ordered_Semigroup | [
"Ordinal Arithmetic"
] | [
"Definition:Ordered Semigroup",
"Definition:Class of All Ordinals",
"Definition:Ordinal Addition"
] | [
"Ordinals under Addition form Semigroup",
"Subset is Compatible with Ordinal Addition",
"Category:Ordinal Arithmetic"
] |
proofwiki-12210 | Subset is Left Compatible with Ordinal Addition | Let $x, y, z$ be ordinals.
Then:
:$x \le y \implies \paren {z + x} \le \paren {z + y}$ | The result follows from Membership is Left Compatible with Ordinal Addition.
{{qed}}
Category:Ordinal Arithmetic
49mwa205l5rwaqgavvl8dr7kwd8c0x8 | Let $x, y, z$ be [[Definition:Ordinal|ordinals]].
Then:
:$x \le y \implies \paren {z + x} \le \paren {z + y}$ | The result follows from [[Membership is Left Compatible with Ordinal Addition]].
{{qed}}
[[Category:Ordinal Arithmetic]]
49mwa205l5rwaqgavvl8dr7kwd8c0x8 | Subset is Left Compatible with Ordinal Addition | https://proofwiki.org/wiki/Subset_is_Left_Compatible_with_Ordinal_Addition | https://proofwiki.org/wiki/Subset_is_Left_Compatible_with_Ordinal_Addition | [
"Ordinal Arithmetic"
] | [
"Definition:Ordinal"
] | [
"Membership is Left Compatible with Ordinal Addition",
"Category:Ordinal Arithmetic"
] |
proofwiki-12211 | Subset is Compatible with Ordinal Addition | Let $x, y, z$ be ordinals.
Then:
:$(1): \quad x \le y \implies \paren {z + x} \le \paren {z + y}$
:$(2): \quad x \le y \implies \paren {x + z} \le \paren {y + z}$ | The result follows from Subset is Left Compatible with Ordinal Addition and Subset is Right Compatible with Ordinal Addition.
{{qed}}
Category:Ordinal Arithmetic
tnh5ow7h12m60x4xi6nsx7h21p49lkp | Let $x, y, z$ be [[Definition:Ordinal|ordinals]].
Then:
:$(1): \quad x \le y \implies \paren {z + x} \le \paren {z + y}$
:$(2): \quad x \le y \implies \paren {x + z} \le \paren {y + z}$ | The result follows from [[Subset is Left Compatible with Ordinal Addition]] and [[Subset is Right Compatible with Ordinal Addition]].
{{qed}}
[[Category:Ordinal Arithmetic]]
tnh5ow7h12m60x4xi6nsx7h21p49lkp | Subset is Compatible with Ordinal Addition | https://proofwiki.org/wiki/Subset_is_Compatible_with_Ordinal_Addition | https://proofwiki.org/wiki/Subset_is_Compatible_with_Ordinal_Addition | [
"Ordinal Arithmetic"
] | [
"Definition:Ordinal"
] | [
"Subset is Left Compatible with Ordinal Addition",
"Subset is Right Compatible with Ordinal Addition",
"Category:Ordinal Arithmetic"
] |
proofwiki-12212 | Ordinals under Addition form Semigroup | $\struct {\On, +}$ forms an semigroup, where:
:$\On$ denotes the class of all ordinals
:$+$ denotes ordinal addition. | The result follows from Ordinal Addition is Closed and Ordinal Addition is Associative.
{{qed}}
Category:Ordinal Arithmetic
jnv27w4ulw2yj7bss9t1hpvl92khmel | $\struct {\On, +}$ forms an [[Definition:Semigroup|semigroup]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$+$ denotes [[Definition:Ordinal Addition|ordinal addition]]. | The result follows from [[Ordinal Addition is Closed]] and [[Ordinal Addition is Associative]].
{{qed}}
[[Category:Ordinal Arithmetic]]
jnv27w4ulw2yj7bss9t1hpvl92khmel | Ordinals under Addition form Semigroup | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Semigroup | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Semigroup | [
"Ordinal Arithmetic"
] | [
"Definition:Semigroup",
"Definition:Class of All Ordinals",
"Definition:Ordinal Addition"
] | [
"Ordinal Addition is Closed",
"Ordinal Addition is Associative",
"Category:Ordinal Arithmetic"
] |
proofwiki-12213 | Ordinals under Multiplication form Semigroup | $\struct {\On, \times}$ forms an semigroup, where:
:$\On$ denotes the class of all ordinals
:$\times$ denotes ordinal multiplication. | The result follows from Ordinal Multiplication is Closed and Ordinal Multiplication is Associative.
{{qed}}
Category:Ordinal Arithmetic
3ah3i1ewrt3oypxxa65nzung54kfen3 | $\struct {\On, \times}$ forms an [[Definition:Semigroup|semigroup]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$\times$ denotes [[Definition:Ordinal Multiplication|ordinal multiplication]]. | The result follows from [[Ordinal Multiplication is Closed]] and [[Ordinal Multiplication is Associative]].
{{qed}}
[[Category:Ordinal Arithmetic]]
3ah3i1ewrt3oypxxa65nzung54kfen3 | Ordinals under Multiplication form Semigroup | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Semigroup | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Semigroup | [
"Ordinal Arithmetic"
] | [
"Definition:Semigroup",
"Definition:Class of All Ordinals",
"Definition:Ordinal Multiplication"
] | [
"Ordinal Multiplication is Closed",
"Ordinal Multiplication is Associative",
"Category:Ordinal Arithmetic"
] |
proofwiki-12214 | Subset is Left Compatible with Ordinal Multiplication | Let $x, y, z$ be ordinals.
Then:
:$x \le y \implies \paren {z \cdot x} \le \paren {z \cdot y}$ | The result follows from Membership is Left Compatible with Ordinal Multiplication.
{{qed}}
Category:Ordinal Arithmetic
nbbl4noxedxydteko8x3i0py1q6rquy | Let $x, y, z$ be [[Definition:Ordinal|ordinals]].
Then:
:$x \le y \implies \paren {z \cdot x} \le \paren {z \cdot y}$ | The result follows from [[Membership is Left Compatible with Ordinal Multiplication]].
{{qed}}
[[Category:Ordinal Arithmetic]]
nbbl4noxedxydteko8x3i0py1q6rquy | Subset is Left Compatible with Ordinal Multiplication | https://proofwiki.org/wiki/Subset_is_Left_Compatible_with_Ordinal_Multiplication | https://proofwiki.org/wiki/Subset_is_Left_Compatible_with_Ordinal_Multiplication | [
"Ordinal Arithmetic"
] | [
"Definition:Ordinal"
] | [
"Membership is Left Compatible with Ordinal Multiplication",
"Category:Ordinal Arithmetic"
] |
proofwiki-12215 | Subset is Compatible with Ordinal Multiplication | Let $x, y, z$ be ordinals.
Then:
:$(1): \quad x \le y \implies \paren {z \cdot x} \le \paren {z \cdot y}$
:$(2): \quad x \le y \implies \paren {x \cdot z} \le \paren {y \cdot z}$ | The result follows from Subset is Left Compatible with Ordinal Multiplication and Subset is Right Compatible with Ordinal Multiplication.
{{qed}}
Category:Ordinal Arithmetic
bgxr6qjjz1yiwwtldsoeknzd90zjzpd | Let $x, y, z$ be [[Definition:Ordinal|ordinals]].
Then:
:$(1): \quad x \le y \implies \paren {z \cdot x} \le \paren {z \cdot y}$
:$(2): \quad x \le y \implies \paren {x \cdot z} \le \paren {y \cdot z}$ | The result follows from [[Subset is Left Compatible with Ordinal Multiplication]] and [[Subset is Right Compatible with Ordinal Multiplication]].
{{qed}}
[[Category:Ordinal Arithmetic]]
bgxr6qjjz1yiwwtldsoeknzd90zjzpd | Subset is Compatible with Ordinal Multiplication | https://proofwiki.org/wiki/Subset_is_Compatible_with_Ordinal_Multiplication | https://proofwiki.org/wiki/Subset_is_Compatible_with_Ordinal_Multiplication | [
"Ordinal Arithmetic"
] | [
"Definition:Ordinal"
] | [
"Subset is Left Compatible with Ordinal Multiplication",
"Subset is Right Compatible with Ordinal Multiplication",
"Category:Ordinal Arithmetic"
] |
proofwiki-12216 | Ordinals under Multiplication form Ordered Semigroup | $\struct {\On, \times, \le}$ forms an ordered semigroup, where:
:$\On$ denotes the class of all ordinals
:$\times$ denotes ordinal multiplication. | The result follows from Ordinals under Multiplication form Semigroup and Subset is Compatible with Ordinal Multiplication.
{{qed}}
Category:Ordinal Arithmetic
hrhb6j3x90u2f8z2aytlnh30sc8eeyq | $\struct {\On, \times, \le}$ forms an [[Definition:Ordered Semigroup|ordered semigroup]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$\times$ denotes [[Definition:Ordinal Multiplication|ordinal multiplication]]. | The result follows from [[Ordinals under Multiplication form Semigroup]] and [[Subset is Compatible with Ordinal Multiplication]].
{{qed}}
[[Category:Ordinal Arithmetic]]
hrhb6j3x90u2f8z2aytlnh30sc8eeyq | Ordinals under Multiplication form Ordered Semigroup | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Ordered_Semigroup | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Ordered_Semigroup | [
"Ordinal Arithmetic"
] | [
"Definition:Ordered Semigroup",
"Definition:Class of All Ordinals",
"Definition:Ordinal Multiplication"
] | [
"Ordinals under Multiplication form Semigroup",
"Subset is Compatible with Ordinal Multiplication",
"Category:Ordinal Arithmetic"
] |
proofwiki-12217 | Ordinals under Addition form Monoid | $\struct {\On, +}$ forms an monoid, where:
:$\On$ denotes the class of all ordinals
:$+$ denotes ordinal addition. | The result follows from Ordinals under Addition form Semigroup and Ordinal Addition by Zero.
{{qed}}
Category:Ordinal Arithmetic
3u2ecmtz643ai87qqd4p7eh1jdq5q43 | $\struct {\On, +}$ forms an [[Definition:Monoid|monoid]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$+$ denotes [[Definition:Ordinal Addition|ordinal addition]]. | The result follows from [[Ordinals under Addition form Semigroup]] and [[Ordinal Addition by Zero]].
{{qed}}
[[Category:Ordinal Arithmetic]]
3u2ecmtz643ai87qqd4p7eh1jdq5q43 | Ordinals under Addition form Monoid | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Monoid | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Monoid | [
"Ordinal Arithmetic"
] | [
"Definition:Monoid",
"Definition:Class of All Ordinals",
"Definition:Ordinal Addition"
] | [
"Ordinals under Addition form Semigroup",
"Ordinal Addition by Zero",
"Category:Ordinal Arithmetic"
] |
proofwiki-12218 | Ordinals under Addition form Ordered Monoid | $\struct {\On, +, \le}$ forms an ordered monoid, where:
:$\On$ denotes the class of all ordinals
:$+$ denotes ordinal addition. | The result follows from Ordinals under Addition form Monoid and Ordinals under Addition form Ordered Semigroup.
{{qed}}
Category:Ordinal Arithmetic
pyuq3l2vwaucge2hh0q77jppigcjlpw | $\struct {\On, +, \le}$ forms an [[Definition:Ordered Monoid|ordered monoid]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$+$ denotes [[Definition:Ordinal Addition|ordinal addition]]. | The result follows from [[Ordinals under Addition form Monoid]] and [[Ordinals under Addition form Ordered Semigroup]].
{{qed}}
[[Category:Ordinal Arithmetic]]
pyuq3l2vwaucge2hh0q77jppigcjlpw | Ordinals under Addition form Ordered Monoid | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Ordered_Monoid | https://proofwiki.org/wiki/Ordinals_under_Addition_form_Ordered_Monoid | [
"Ordinal Arithmetic"
] | [
"Definition:Ordered Monoid",
"Definition:Class of All Ordinals",
"Definition:Ordinal Addition"
] | [
"Ordinals under Addition form Monoid",
"Ordinals under Addition form Ordered Semigroup",
"Category:Ordinal Arithmetic"
] |
proofwiki-12219 | Ordinals under Multiplication form Monoid | $\struct {\On, \times}$ forms an monoid, where:
:$\On$ denotes the class of all ordinals
:$\times$ denotes ordinal multiplication. | The result follows from Ordinals under Multiplication form Semigroup and Ordinal Multiplication by One.
{{qed}}
Category:Ordinal Arithmetic
qf2fhfd9q8s8z4hh3vizhegcpnuvut1 | $\struct {\On, \times}$ forms an [[Definition:Monoid|monoid]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$\times$ denotes [[Definition:Ordinal Multiplication|ordinal multiplication]]. | The result follows from [[Ordinals under Multiplication form Semigroup]] and [[Ordinal Multiplication by One]].
{{qed}}
[[Category:Ordinal Arithmetic]]
qf2fhfd9q8s8z4hh3vizhegcpnuvut1 | Ordinals under Multiplication form Monoid | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Monoid | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Monoid | [
"Ordinal Arithmetic"
] | [
"Definition:Monoid",
"Definition:Class of All Ordinals",
"Definition:Ordinal Multiplication"
] | [
"Ordinals under Multiplication form Semigroup",
"Ordinal Multiplication by One",
"Category:Ordinal Arithmetic"
] |
proofwiki-12220 | Ordinals under Multiplication form Ordered Monoid | $\struct {\On, \times, \le}$ forms an ordered monoid, where:
:$\On$ denotes the class of all ordinals
:$\times$ denotes ordinal multiplication. | The result follows from Ordinals under Multiplication form Monoid and Ordinals under Multiplication form Ordered Semigroup.
{{qed}}
Category:Ordinal Arithmetic
3frch88jn92dtba6d2q09fwm97dzfqy | $\struct {\On, \times, \le}$ forms an [[Definition:Ordered Monoid|ordered monoid]], where:
:$\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]
:$\times$ denotes [[Definition:Ordinal Multiplication|ordinal multiplication]]. | The result follows from [[Ordinals under Multiplication form Monoid]] and [[Ordinals under Multiplication form Ordered Semigroup]].
{{qed}}
[[Category:Ordinal Arithmetic]]
3frch88jn92dtba6d2q09fwm97dzfqy | Ordinals under Multiplication form Ordered Monoid | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Ordered_Monoid | https://proofwiki.org/wiki/Ordinals_under_Multiplication_form_Ordered_Monoid | [
"Ordinal Arithmetic"
] | [
"Definition:Ordered Monoid",
"Definition:Class of All Ordinals",
"Definition:Ordinal Multiplication"
] | [
"Ordinals under Multiplication form Monoid",
"Ordinals under Multiplication form Ordered Semigroup",
"Category:Ordinal Arithmetic"
] |
proofwiki-12221 | Buffon's Needle | Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards.
Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height.
Then the probability that the needle falls across one of the parallel li... | Let $N$ refer to the needle.
For simplicity, consider the real number plane $\R^2$ divided into strips by the lines $x = k$ for each integer $k$.
Then $N$ would have length $1$, which is the distance between the lines.
Define $\theta \in \hointr {-\dfrac \pi 2} {\dfrac \pi 2}$ as the angle between $N$ and the $x$-axis.... | Let a [[Definition:Horizontal|horizontal]] [[Definition:Plane|plane]] be divided into strips by a series of [[Definition:Parallel Lines|parallel lines]] a fixed [[Definition:Distance between Parallel Lines|distance]] apart, like floorboards.
Let a needle whose [[Definition:Length of Line|length]] equals the [[Definiti... | Let $N$ refer to the needle.
For simplicity, consider the [[Definition:Real Number Plane|real number plane $\R^2$]] divided into strips by the lines $x = k$ for each [[Definition:Integer|integer]] $k$.
Then $N$ would have [[Definition:Length of Line|length]] $1$, which is the [[Definition:Distance between Parallel Li... | Buffon's Needle | https://proofwiki.org/wiki/Buffon's_Needle | https://proofwiki.org/wiki/Buffon's_Needle | [
"Buffon's Needle",
"Trigonometry",
"Pi"
] | [
"Definition:Horizontal",
"Definition:Plane Surface",
"Definition:Parallel (Geometry)/Lines",
"Definition:Distance between Parallel Lines",
"Definition:Linear Measure/Length",
"Definition:Distance between Parallel Lines",
"Definition:Parallel (Geometry)/Lines",
"Definition:Plane Surface",
"Definition... | [
"Definition:Real Number Plane",
"Definition:Integer",
"Definition:Linear Measure/Length",
"Definition:Distance between Parallel Lines",
"Definition:Parallel (Geometry)/Lines",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Horizontal",
"Definition:Component",
"Definition:Linear Measure/... |
proofwiki-12222 | Pi as Sum of Odd Reciprocals Alternating in Sign in Pairs | :$\dfrac {\pi \sqrt 2} 4 = 1 + \dfrac 1 3 - \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} - \dfrac 1 {13} - \dfrac 1 {15} \cdots$ | Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = \dfrac {x^1} 1 + \dfrac {x^3} 3 - \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 + \dfrac {x^{11} } {11} - \dfrac {x^{13} } {13} - \dfrac {x^{15} } {15} \cdots$
We first confirm that the series will converge at $x = 1$.
By grouping the... | :$\dfrac {\pi \sqrt 2} 4 = 1 + \dfrac 1 3 - \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} - \dfrac 1 {13} - \dfrac 1 {15} \cdots$ | Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map f x = \dfrac {x^1} 1 + \dfrac {x^3} 3 - \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 + \dfrac {x^{11} } {11} - \dfrac {x^{13} } {13} - \dfrac {x^{15} } {15} \cdots$
We first confirm that the [[Definition:Rea... | Pi as Sum of Odd Reciprocals Alternating in Sign in Pairs | https://proofwiki.org/wiki/Pi_as_Sum_of_Odd_Reciprocals_Alternating_in_Sign_in_Pairs | https://proofwiki.org/wiki/Pi_as_Sum_of_Odd_Reciprocals_Alternating_in_Sign_in_Pairs | [
"Formulas for Pi"
] | [] | [
"Definition:Real Function",
"Definition:Series/Real",
"Definition:Convergent Series/Number Field",
"Definition:Series/Real",
"Definition:Term of Sequence",
"Definition:Convergent Series/Number Field",
"Alternating Series Test",
"Power Rule for Derivatives",
"Exponent Combination Laws/Product of Powe... |
proofwiki-12223 | Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals | :$\dfrac {\pi - 3} 4 = \dfrac 1 {2 \times 3 \times 4} - \dfrac 1 {4 \times 5 \times 6} + \dfrac 1 {6 \times 7 \times 8} \cdots$ | The alternating sum can be written as $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1}} {2 n \paren{2 n + 1} \paren{2 n + 2}}$.
By partial fraction decomposition:
:$\ds \frac 1 {2 n \paren {2 n + 1} \paren {2 n + 2}} = \frac 1 2 \paren{\frac 1 {2 n} - \frac 2 {2 n + 1} + \frac 1 {2 n + 2}}$
Therefore:
{{begi... | :$\dfrac {\pi - 3} 4 = \dfrac 1 {2 \times 3 \times 4} - \dfrac 1 {4 \times 5 \times 6} + \dfrac 1 {6 \times 7 \times 8} \cdots$ | The alternating sum can be written as $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1}} {2 n \paren{2 n + 1} \paren{2 n + 2}}$.
By [[Partial Fractions Expansion/Examples/1 over 2x(2x+1)(2x+2)|partial fraction decomposition]]:
:$\ds \frac 1 {2 n \paren {2 n + 1} \paren {2 n + 2}} = \frac 1 2 \paren{\frac 1 ... | Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals/Proof 2 | https://proofwiki.org/wiki/Pi_as_Sum_of_Alternating_Sequence_of_Products_of_3_Consecutive_Reciprocals | https://proofwiki.org/wiki/Pi_as_Sum_of_Alternating_Sequence_of_Products_of_3_Consecutive_Reciprocals/Proof_2 | [
"Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals",
"Formulas for Pi"
] | [] | [
"Partial Fractions Expansion/Examples/1 over 2x(2x+1)(2x+2)",
"Translation of Index Variable of Summation",
"Leibniz's Formula for Pi"
] |
proofwiki-12224 | Basel Problem as Infinite Product | :$\ds \dfrac {\pi^2} 6 = \prod_{p \mathop \in \mathbb P} \dfrac {p^2} {p^2 - 1}$ | From Sum of Reciprocals of Powers as Euler Product:
:$\ds \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$
for $z \in \C$ such that $\map \Re z > 1$.
Putting $z = 2$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \ge 1} \dfrac 1 {n^2}
| r = \prod_p \frac 1 {1 - p^{-2} }
| c =
}}
{{eqn | r... | :$\ds \dfrac {\pi^2} 6 = \prod_{p \mathop \in \mathbb P} \dfrac {p^2} {p^2 - 1}$ | From [[Sum of Reciprocals of Powers as Euler Product]]:
:$\ds \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$
for $z \in \C$ such that $\map \Re z > 1$.
Putting $z = 2$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \ge 1} \dfrac 1 {n^2}
| r = \prod_p \frac 1 {1 - p^{-2} }
| c =
}}
{... | Basel Problem as Infinite Product | https://proofwiki.org/wiki/Basel_Problem_as_Infinite_Product | https://proofwiki.org/wiki/Basel_Problem_as_Infinite_Product | [
"Basel Problem"
] | [] | [
"Sum of Reciprocals of Powers as Euler Product",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Riemann Zeta Function at Even Integers/Examples/2"
] |
proofwiki-12225 | Convergence of Taylor Series of Function Analytic on Disk | Let $F$ be a complex function.
Let $x_0$ be a point in $\R$.
Let $R$ be an extended real number greater than zero.
Let $F$ be analytic at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } < R$
where $\tuple {x_0, 0}$ denotes the complex number with real part $x_0$ and imaginary part $0$.
Let the restricti... | === Lemma ===
{{:Convergence of Taylor Series of Function Analytic on Disk/Lemma}}{{qed|lemma}}
Let $r$ be a real number satisfying:
:$0 < r < R$
Let $x$ be a real number satisfying:
:$\size {x - x_0} < r$
$f$ has a Taylor series expansion about $x_0$ with radius of convergence greater than zero as $f$ is analytic at $... | Let $F$ be a [[Definition:Complex Function|complex function]].
Let $x_0$ be a point in $\R$.
Let $R$ be an [[Definition:Extended Real Number Line|extended real number]] greater than zero.
Let $F$ be [[Definition:Analytic Complex Function|analytic]] at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } <... | === [[Convergence of Taylor Series of Function Analytic on Disk/Lemma|Lemma]] ===
{{:Convergence of Taylor Series of Function Analytic on Disk/Lemma}}{{qed|lemma}}
Let $r$ be a [[Definition:Real Number|real number]] satisfying:
:$0 < r < R$
Let $x$ be a [[Definition:Real Number|real number]] satisfying:
:$\size {x -... | Convergence of Taylor Series of Function Analytic on Disk | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk | [
"Taylor Series",
"Real Analysis"
] | [
"Definition:Complex Function",
"Definition:Extended Real Number Line",
"Definition:Analytic Function/Complex Plane",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Restriction/Mapping",
"Definition:Real Function",
"Definiti... | [
"Convergence of Taylor Series of Function Analytic on Disk/Lemma",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Taylor Series",
"Definition:Radius of Convergence/Real Domain",
"Definition:Analytic Function/Real Numbers",
"Definition:Taylor Series",
"Definition:Definite Integral/Limi... |
proofwiki-12226 | Convergence of Taylor Series of Function Analytic on Disk | Let $F$ be a complex function.
Let $x_0$ be a point in $\R$.
Let $R$ be an extended real number greater than zero.
Let $F$ be analytic at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } < R$
where $\tuple {x_0, 0}$ denotes the complex number with real part $x_0$ and imaginary part $0$.
Let the restricti... | Let $z$ be a real number.
We have:
:$\ds \sum_{n \mathop = 0}^\infty n z^n$ has radius of convergence of $1$
by Complex Power Series/Examples/n.
This leads to that:
:$\ds \sum_{n \mathop = 0}^\infty n \paren {\frac 1 y}^n$ converges
as $\ds \frac 1 y < 1$ as $y > 1$.
This leads to that:
:the sequence $\ds \sequence {\f... | Let $F$ be a [[Definition:Complex Function|complex function]].
Let $x_0$ be a point in $\R$.
Let $R$ be an [[Definition:Extended Real Number Line|extended real number]] greater than zero.
Let $F$ be [[Definition:Analytic Complex Function|analytic]] at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } <... | Let $z$ be a [[Definition:Real Number|real number]].
We have:
:$\ds \sum_{n \mathop = 0}^\infty n z^n$ has [[Definition:Radius of Convergence of Real Power Series|radius of convergence]] of $1$
by [[Complex Power Series/Examples/n]].
This leads to that:
:$\ds \sum_{n \mathop = 0}^\infty n \paren {\frac 1 y}^n$ [[Def... | Convergence of Taylor Series of Function Analytic on Disk/Lemma/Proof 1 | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk/Lemma/Proof_1 | [
"Taylor Series",
"Real Analysis"
] | [
"Definition:Complex Function",
"Definition:Extended Real Number Line",
"Definition:Analytic Function/Complex Plane",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Restriction/Mapping",
"Definition:Real Function",
"Definiti... | [
"Definition:Real Number",
"Definition:Radius of Convergence/Real Domain",
"Complex Power Series/Examples/n",
"Definition:Convergent Series/Number Field",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Terms in Convergent Series ... |
proofwiki-12227 | Convergence of Taylor Series of Function Analytic on Disk | Let $F$ be a complex function.
Let $x_0$ be a point in $\R$.
Let $R$ be an extended real number greater than zero.
Let $F$ be analytic at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } < R$
where $\tuple {x_0, 0}$ denotes the complex number with real part $x_0$ and imaginary part $0$.
Let the restricti... | Note that $\ln y > 0$ as $y > 1$.
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac n {y^n}
| r = \lim_{n \mathop \to \infty} \frac n {\paren {e^{\ln y} }^n}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac n {e^{\paren {\ln y} n} }
}}
{{eqn | r = 0
| c = as $\ds \lim_{x \mathop \to \infty} \frac ... | Let $F$ be a [[Definition:Complex Function|complex function]].
Let $x_0$ be a point in $\R$.
Let $R$ be an [[Definition:Extended Real Number Line|extended real number]] greater than zero.
Let $F$ be [[Definition:Analytic Complex Function|analytic]] at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } <... | Note that $\ln y > 0$ as $y > 1$.
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac n {y^n}
| r = \lim_{n \mathop \to \infty} \frac n {\paren {e^{\ln y} }^n}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac n {e^{\paren {\ln y} n} }
}}
{{eqn | r = 0
| c = as $\ds \lim_{x \mathop \to \infty} \frac... | Convergence of Taylor Series of Function Analytic on Disk/Lemma/Proof 2 | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk | https://proofwiki.org/wiki/Convergence_of_Taylor_Series_of_Function_Analytic_on_Disk/Lemma/Proof_2 | [
"Taylor Series",
"Real Analysis"
] | [
"Definition:Complex Function",
"Definition:Extended Real Number Line",
"Definition:Analytic Function/Complex Plane",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Restriction/Mapping",
"Definition:Real Function",
"Definiti... | [
"Limit at Infinity of Polynomial over Complex Exponential"
] |
proofwiki-12228 | Taylor Series of Analytic Function has infinite Radius of Convergence | Let $F$ be a complex function.
Let $F$ be analytic everywhere.
Let the restriction of $F$ to $\R \to \C$ be a real function $f$.
This means:
:$\forall x \in \R: \map f x = \map \Re {\map F {x, 0} }, 0 = \map \Im {\map F {x, 0} }$
where $\tuple {x, 0}$ denotes the complex number with real part $x$ and imaginary part $0$... | The result follows by Convergence of Taylor Series of Function Analytic on Disk for the case $R = \infty$.
{{qed}} | Let $F$ be a [[Definition:Complex Function|complex function]].
Let $F$ be [[Definition:Analytic Complex Function|analytic]] everywhere.
Let the [[Definition:Restriction of Mapping|restriction]] of $F$ to $\R \to \C$ be a [[Definition:Real Function|real function]] $f$.
This means:
:$\forall x \in \R: \map f x = \map... | The result follows by [[Convergence of Taylor Series of Function Analytic on Disk]] for the case $R = \infty$.
{{qed}} | Taylor Series of Analytic Function has infinite Radius of Convergence | https://proofwiki.org/wiki/Taylor_Series_of_Analytic_Function_has_infinite_Radius_of_Convergence | https://proofwiki.org/wiki/Taylor_Series_of_Analytic_Function_has_infinite_Radius_of_Convergence | [
"Taylor Series",
"Real Analysis"
] | [
"Definition:Complex Function",
"Definition:Analytic Function/Complex Plane",
"Definition:Restriction/Mapping",
"Definition:Real Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Taylor Series",
"Definition:Convergen... | [
"Convergence of Taylor Series of Function Analytic on Disk"
] |
proofwiki-12229 | Taylor Series reaches closest Singularity | Let $F$ be a complex function.
Let $F$ be analytic everywhere except at a finite number of singularities.
Let a singularity of $F$ be one of the following:
:a pole
:an essential singularity
:a branch point
In the latter case $F$ is a restriction of a multifunction to one of its branches.
Let $x_0$ be a real number.
Let... | We have that $F$ is analytic everywhere except at its singularities.
Also, the distance from the complex number $\tuple {x_0, 0}$ to the closest singularity of $F$ is $R$.
Therefore:
:$F$ is analytic at every point $z \in \C$ satisfying $\size {z - \tuple {x_0, 0} } < R$
where $\tuple {x_0 , 0}$ denotes the complex num... | Let $F$ be a [[Definition:Complex Function|complex function]].
Let $F$ be [[Definition:Analytic Complex Function|analytic]] everywhere except at a [[Definition:Finite Set|finite]] number of singularities.
Let a singularity of $F$ be one of the following:
:a [[Definition:Pole (Complex Analysis)|pole]]
:an [[Definition... | We have that $F$ is [[Definition:Analytic Complex Function|analytic]] everywhere except at its singularities.
Also, the [[Definition:Distance between Complex Numbers|distance]] from the [[Definition:Complex Number|complex number]] $\tuple {x_0, 0}$ to the closest singularity of $F$ is $R$.
Therefore:
:$F$ is [[Defini... | Taylor Series reaches closest Singularity | https://proofwiki.org/wiki/Taylor_Series_reaches_closest_Singularity | https://proofwiki.org/wiki/Taylor_Series_reaches_closest_Singularity | [
"Taylor Series",
"Real Analysis"
] | [
"Definition:Complex Function",
"Definition:Analytic Function/Complex Plane",
"Definition:Finite Set",
"Definition:Isolated Singularity/Pole",
"Definition:Isolated Singularity",
"Definition:Branch Point of Complex Multifunction",
"Definition:Restriction/Mapping",
"Definition:Left-Total Relation/Multifu... | [
"Definition:Analytic Function/Complex Plane",
"Definition:Distance/Points/Complex Numbers",
"Definition:Complex Number",
"Definition:Analytic Function/Complex Plane",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Convergence of Taylor ... |
proofwiki-12230 | Number of Binary Digits in Power of 10 | Let $n$ be a positive integer.
Expressed in binary notation, the number of digits in the $n$th power of $10$:
:$10^n$
is equal to:
:$\ceiling {n \log_2 10}$
where $\ceiling x$ denotes the ceiling of $x$. | Let $10^n$ have $m$ digits when expressed in binary notation.
By the Basis Representation Theorem and its implications, a positive integer $x$ has $m$ digits {{iff}}:
:$2^{m - 1} \le x < 2^m$
Thus:
{{begin-eqn}}
{{eqn | l = 2^{m - 1}
| o = \le
| m = 10^n
| mo= <
| r = 2^m
| c =
}}
{{eqn |... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Expressed in [[Definition:Binary Notation|binary notation]], the number of [[Definition:Digit|digits]] in the [[Definition:Integer Power|$n$th power]] of $10$:
:$10^n$
is equal to:
:$\ceiling {n \log_2 10}$
where $\ceiling x$ denotes the [[Definition:Ceili... | Let $10^n$ have $m$ [[Definition:Digit|digits]] when expressed in [[Definition:Binary Notation|binary notation]].
By the [[Basis Representation Theorem]] and its implications, a [[Definition:Positive Integer|positive integer]] $x$ has $m$ [[Definition:Digit|digits]] {{iff}}:
:$2^{m - 1} \le x < 2^m$
Thus:
{{begin-eqn... | Number of Binary Digits in Power of 10 | https://proofwiki.org/wiki/Number_of_Binary_Digits_in_Power_of_10 | https://proofwiki.org/wiki/Number_of_Binary_Digits_in_Power_of_10 | [
"Powers of 10",
"Logarithms",
"Number of Binary Digits in Power of 10"
] | [
"Definition:Positive/Integer",
"Definition:Binary Notation",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Ceiling Function"
] | [
"Definition:Digit",
"Definition:Binary Notation",
"Basis Representation Theorem",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Integer equals Ceiling iff Number between Integer and One Less"
] |
proofwiki-12231 | Number of Binary Digits in Power of 10/Example/1000 | When expressed in binary notation, the number of digits in $1000$ is $10$. | Let $m$ be the number of digits in $1000$.
From Number of Binary Digits in Power of 10:
:$m = \ceiling {3 \log_2 10}$
From Logarithm Base 2 of 10:
:$\log_2 10 \approx 3 \cdotp 32192 \, 8 \ldots$
and so:
:$m \approx 9 \cdotp 96$
Hence the result.
The actual number is:
:$1000_{10} = 1 \, 111 \, 101 \, 100_2$
{{qed}} | When expressed in [[Definition:Binary Notation|binary notation]], the number of [[Definition:Digit|digits]] in $1000$ is $10$. | Let $m$ be the number of [[Definition:Digit|digits]] in $1000$.
From [[Number of Binary Digits in Power of 10]]:
:$m = \ceiling {3 \log_2 10}$
From [[Logarithm Base 2 of 10]]:
:$\log_2 10 \approx 3 \cdotp 32192 \, 8 \ldots$
and so:
:$m \approx 9 \cdotp 96$
Hence the result.
The actual number is:
:$1000_{10} = 1 \,... | Number of Binary Digits in Power of 10/Example/1000 | https://proofwiki.org/wiki/Number_of_Binary_Digits_in_Power_of_10/Example/1000 | https://proofwiki.org/wiki/Number_of_Binary_Digits_in_Power_of_10/Example/1000 | [
"1000"
] | [
"Definition:Binary Notation",
"Definition:Digit"
] | [
"Definition:Digit",
"Number of Binary Digits in Power of 10",
"Binary Logarithm/Examples/10"
] |
proofwiki-12232 | Bound for Analytic Function and Derivatives | Let $f$ be a complex function.
Let $z_0$ be a point in $\C$.
Let $r$ be a real number in $\R_{>0}$.
Let $\Gamma$ be a circle in $\C$ with center at $z_0$ and radius $r$.
Let $f$ be analytic on $\Gamma$ and its interior.
Let $t \in \C$ be such that $\cmod {t - z_0} < r$.
Then a real number $M$ exists such that, for ever... | === Lemma (Analytic Function Bounded on Circle) ===
{{:Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle}}{{qed|lemma}}
We have that $f$ is bounded on $\Gamma$ by Lemma (Analytic Function Bounded on Circle).
Therefore, there is a positive real number $M$ that satisfies:
:$(1): \quad \foral... | Let $f$ be a [[Definition:Complex Function|complex function]].
Let $z_0$ be a point in $\C$.
Let $r$ be a [[Definition:Real Number|real number]] in $\R_{>0}$.
Let $\Gamma$ be a [[Definition:Circle|circle]] in $\C$ with [[Definition:Center of Circle|center]] at $z_0$ and [[Definition:Radius of Circle|radius]] $r$.
L... | === [[Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle|Lemma (Analytic Function Bounded on Circle)]] ===
{{:Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle}}{{qed|lemma}}
We have that $f$ is [[Definition:Bounded Complex-Valued Function|bounded]] on $\Gamm... | Bound for Analytic Function and Derivatives | https://proofwiki.org/wiki/Bound_for_Analytic_Function_and_Derivatives | https://proofwiki.org/wiki/Bound_for_Analytic_Function_and_Derivatives | [
"Analytic Complex Functions"
] | [
"Definition:Complex Function",
"Definition:Real Number",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Analytic Function/Complex Plane",
"Definition:Jordan Curve/Interior",
"Definition:Positive/Real Number"
] | [
"Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle",
"Definition:Bounded Mapping/Complex-Valued",
"Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle",
"Definition:Positive/Real Number",
"Definition:Analytic Function/Complex Plane",
"Definition:Jo... |
proofwiki-12233 | Regular Octahedron is Dual of Cube | The regular octahedron is the dual of the cube. | :300px
{{ProofWanted}} | The [[Definition:Regular Octahedron|regular octahedron]] is the [[Definition:Dual Polyhedron|dual]] of the [[Definition:Cube (Geometry)|cube]]. | :[[File:DualCubeOctahedron.png|300px]]
{{ProofWanted}} | Regular Octahedron is Dual of Cube | https://proofwiki.org/wiki/Regular_Octahedron_is_Dual_of_Cube | https://proofwiki.org/wiki/Regular_Octahedron_is_Dual_of_Cube | [
"Regular Octahedra",
"Cubes"
] | [
"Definition:Octahedron/Regular",
"Definition:Dual Polyhedron",
"Definition:Cube/Geometry"
] | [
"File:DualCubeOctahedron.png"
] |
proofwiki-12234 | Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts | Let $F$ be a plane figure.
Let $F$ have two different axes of bilateral symmetry.
Then those two axes divide $F$ into $4$ congruent parts.
:500px | {{ProofWanted|More background needed on the geometry of symmetry.}} | Let $F$ be a [[Definition:Plane Figure|plane figure]].
Let $F$ have two different [[Definition:Axis of Bilateral Symmetry|axes of bilateral symmetry]].
Then those two [[Definition:Axis of Bilateral Symmetry|axes]] divide $F$ into $4$ [[Definition:Congruence (Geometry)|congruent]] parts.
:[[File:DoubleBilateralSymmet... | {{ProofWanted|More background needed on the geometry of symmetry.}} | Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts | https://proofwiki.org/wiki/Plane_Figure_with_Bilateral_Symmetry_about_Two_Lines_has_4_Congruent_Parts | https://proofwiki.org/wiki/Plane_Figure_with_Bilateral_Symmetry_about_Two_Lines_has_4_Congruent_Parts | [
"Bilateral Symmetry"
] | [
"Definition:Geometric Figure/Plane Figure",
"Definition:Bilateral Symmetry/Axis",
"Definition:Bilateral Symmetry/Axis",
"Definition:Congruence (Geometry)",
"File:DoubleBilateralSymmetry.png"
] | [] |
proofwiki-12235 | Divisibility of n-1 Factorial by Composite n | Let $n \in \Z$ be composite.
Then:
:$n \divides \paren {n - 1}! \iff n \ne 4$
where:
:$\divides$ denotes divisibility
:$n!$ denotes the factorial of $n$. | === Necessary Condition ===
We have that $3! = 6$ and that $4$ does not divide $6$.
So in order for $n$ to divide $\paren {n - 1}!$ it is necessary that $n \ne 4$.
{{qed|lemma}} | Let $n \in \Z$ be [[Definition:Composite Number|composite]].
Then:
:$n \divides \paren {n - 1}! \iff n \ne 4$
where:
:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
:$n!$ denotes the [[Definition:Factorial|factorial]] of $n$. | === Necessary Condition ===
We have that $3! = 6$ and that $4$ does not [[Definition:Divisor of Integer|divide]] $6$.
So in order for $n$ to [[Definition:Divisor of Integer|divide]] $\paren {n - 1}!$ it is [[Definition:Necessary Condition|necessary]] that $n \ne 4$.
{{qed|lemma}} | Divisibility of n-1 Factorial by Composite n | https://proofwiki.org/wiki/Divisibility_of_n-1_Factorial_by_Composite_n | https://proofwiki.org/wiki/Divisibility_of_n-1_Factorial_by_Composite_n | [
"Factorials"
] | [
"Definition:Composite Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Factorial"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Conditional/Necessary Condition"
] |
proofwiki-12236 | Divisibility by 4 | An integer $N$ expressed in decimal notation is divisible by $4$ {{iff}} the $2$ {{LSD}}s of $N$ form a $2$-digit integer divisible by $4$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $4$
{{iff}}:
:$10 a_1 + a_0$ is divisible by $4$. | Let $N$ be divisible by $4$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 4
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 4
}}
{{eqn | ll= \leadstoandfrom
| l = 10 a_1 + a_0 + 10^2 \sum_{k \mathop = 2}^n a_k 10^{k - 2}... | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $4$ {{iff}} the $2$ {{LSD}}s of $N$ form a $2$-[[Definition:Digit|digit]] [[Definition:Integer|integer]] [[Definition:Divisor of Integer|divisible]] by $4$.
That is:
:$... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $4$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 4
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 4
}}
{{eqn | ll= \leadstoandfrom
| l = 10 a_1 + a_0 + 10^2 \s... | Divisibility by 4 | https://proofwiki.org/wiki/Divisibility_by_4 | https://proofwiki.org/wiki/Divisibility_by_4 | [
"Divisibility Tests",
"4"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12237 | Ratio of Number to Reversal which is Multiple | Take a (strictly) positive integer $n$, written in conventional decimal notation.
Let $m$ be the reversal of $n$.
Let $m = k n$ where $k$ is an integer.
Then $k$ is either $4$ or $9$. | === Existence ===
$8712 = 4 \times 2178$
$9801 = 9 \times 1089$ | Take a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$, written in conventional [[Definition:Decimal Notation|decimal notation]].
Let $m$ be the [[Definition:Reversal|reversal]] of $n$.
Let $m = k n$ where $k$ is an [[Definition:Integer|integer]].
Then $k$ is either $4$ or $9$. | === Existence ===
$8712 = 4 \times 2178$
$9801 = 9 \times 1089$ | Ratio of Number to Reversal which is Multiple | https://proofwiki.org/wiki/Ratio_of_Number_to_Reversal_which_is_Multiple | https://proofwiki.org/wiki/Ratio_of_Number_to_Reversal_which_is_Multiple | [
"Recreational Mathematics",
"Reversals",
"1089",
"2178"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Decimal Notation",
"Definition:Reversal",
"Definition:Integer"
] | [] |
proofwiki-12238 | Smallest Pythagorean Triangle is 3-4-5 | The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$.
:300px | From Solutions of Pythagorean Equation, all Pythagorean triangles, the set of all primitive Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are (strictly) positive integers
:$m \perp n$, that is, $m$ and $n$ are coprime
:$m$ and $n$ are of opposite parity
:$m > n$... | The smallest [[Definition:Pythagorean Triangle|Pythagorean triangle]] has [[Definition:Side of Polygon|sides]] of [[Definition:Length of Line|length]] [[Pythagorean Triangle/Examples/3-4-5|$3$, $4$ and $5$]].
:[[File:3-4-5.png|300px]] | From [[Solutions of Pythagorean Equation]], all [[Definition:Pythagorean Triangle|Pythagorean triangles]], the [[Definition:Set|set]] of all [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triples]] is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are [[Definition:Str... | Smallest Pythagorean Triangle is 3-4-5 | https://proofwiki.org/wiki/Smallest_Pythagorean_Triangle_is_3-4-5 | https://proofwiki.org/wiki/Smallest_Pythagorean_Triangle_is_3-4-5 | [
"Pythagorean Triangles"
] | [
"Definition:Pythagorean Triangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Pythagorean Triangle/Examples/3-4-5",
"File:3-4-5.png"
] | [
"Solutions of Pythagorean Equation",
"Definition:Pythagorean Triangle",
"Definition:Set",
"Definition:Pythagorean Triple/Primitive",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Parity of Integer",
"Definition:Strictly Positive/Integer"
] |
proofwiki-12239 | Pythagorean Triangle with Sides in Arithmetic Sequence | The $3-4-5$ triangle is the only Pythagorean triangle such that:
:the lengths of whose sides are in arithmetic sequence
and:
:the lengths of whose sides form a primitive Pythagorean triple. | Let $a, b, c$ be the lengths of the sides of a Pythagorean triangle such that $a < b < c$.
Let $a, b, c$ be in arithmetic sequence:
:$b - a = c - b$
Let $a, b, c$ form a primitive Pythagorean triple:
:$a \perp b$
By definition of primitive Pythagorean triple, $a, b, c$ are in the form:
:$2 m n, m^2 - n^2, m^2 + n^2$
We... | The [[Pythagorean Triangle/Examples/3-4-5|$3-4-5$ triangle]] is the only [[Definition:Pythagorean Triangle|Pythagorean triangle]] such that:
:the [[Definition:Length of Line|lengths]] of whose [[Definition:Side of Polygon|sides]] are in [[Definition:Arithmetic Sequence|arithmetic sequence]]
and:
:the [[Definition:Lengt... | Let $a, b, c$ be the [[Definition:Length of Line|lengths]] of the [[Definition:Side of Polygon|sides]] of a [[Definition:Pythagorean Triangle|Pythagorean triangle]] such that $a < b < c$.
Let $a, b, c$ be in [[Definition:Arithmetic Sequence|arithmetic sequence]]:
:$b - a = c - b$
Let $a, b, c$ form a [[Definition:Pri... | Pythagorean Triangle with Sides in Arithmetic Sequence | https://proofwiki.org/wiki/Pythagorean_Triangle_with_Sides_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Pythagorean_Triangle_with_Sides_in_Arithmetic_Sequence | [
"Pythagorean Triangles"
] | [
"Pythagorean Triangle/Examples/3-4-5",
"Definition:Pythagorean Triangle",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Arithmetic Sequence",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Pythagorean Triple/Primitive"
] | [
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Pythagorean Triangle",
"Definition:Arithmetic Sequence",
"Definition:Pythagorean Triple/Primitive",
"Definition:Pythagorean Triple/Primitive",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Solutions of Pythagorean ... |
proofwiki-12240 | Injectivity of Laplace Transform | Let $f$, $g$ be functions from $\R_{\ge 0} \to K$ of a real variable $t$, where $K \in \set {\R, \C}$.
Further let $f$ and $g$ be continuous everywhere on their domains.
Let $f$ and $g$ both admit Laplace transforms.
Suppose that the Laplace transforms $\laptrans f$ and $\laptrans g$ satisfy:
:$\forall t \in \R_{\ge 0}... | {{ProofWanted}}
Category:Injectivity of Laplace Transform
Category:Laplace Transforms
31tsnqyswld6dmy5qw1bm23lx3pdbnd | Let $f$, $g$ be [[Definition:Function|functions]] from $\R_{\ge 0} \to K$ of a [[Definition:Independent Variable|real variable]] $t$, where $K \in \set {\R, \C}$.
Further let $f$ and $g$ be [[Definition:Continuity|continuous]] everywhere on their [[Definition:Domain of Mapping|domains]].
Let $f$ and $g$ both admit [[... | {{ProofWanted}}
[[Category:Injectivity of Laplace Transform]]
[[Category:Laplace Transforms]]
31tsnqyswld6dmy5qw1bm23lx3pdbnd | Injectivity of Laplace Transform | https://proofwiki.org/wiki/Injectivity_of_Laplace_Transform | https://proofwiki.org/wiki/Injectivity_of_Laplace_Transform | [
"Injectivity of Laplace Transform",
"Laplace Transforms"
] | [
"Definition:Function",
"Definition:Independent Variable",
"Definition:Continuous",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Laplace Transform"
] | [
"Category:Injectivity of Laplace Transform",
"Category:Laplace Transforms"
] |
proofwiki-12241 | Pythagorean Triangle whose Area is Half Perimeter | The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter. | Let $a, b, c$ be the lengths of the sides of a Pythagorean triangle $T$.
Thus $a, b, c$ form a Pythagorean triple.
By definition of Pythagorean triple, $a, b, c$ are in the form:
:$2 m n, m^2 - n^2, m^2 + n^2$
We have that $m^2 + n^2$ is always the hypotenuse.
Thus the area of $T$ is given by:
:$\AA = m n \paren {m^2 -... | The [[Pythagorean Triangle/Examples/3-4-5|$3-4-5$ triangle]] is the only [[Definition:Pythagorean Triangle|Pythagorean triangle]] whose [[Definition:Area|area]] is half its [[Definition:Perimeter|perimeter]]. | Let $a, b, c$ be the [[Definition:Length of Line|lengths]] of the [[Definition:Side of Polygon|sides]] of a [[Definition:Pythagorean Triangle|Pythagorean triangle]] $T$.
Thus $a, b, c$ form a [[Definition:Pythagorean Triple|Pythagorean triple]].
By definition of [[Definition:Pythagorean Triple|Pythagorean triple]], $... | Pythagorean Triangle whose Area is Half Perimeter | https://proofwiki.org/wiki/Pythagorean_Triangle_whose_Area_is_Half_Perimeter | https://proofwiki.org/wiki/Pythagorean_Triangle_whose_Area_is_Half_Perimeter | [
"Pythagorean Triangles"
] | [
"Pythagorean Triangle/Examples/3-4-5",
"Definition:Pythagorean Triangle",
"Definition:Area",
"Definition:Perimeter"
] | [
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Pythagorean Triangle",
"Definition:Pythagorean Triple",
"Definition:Pythagorean Triple",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Area",
"Definition:Perimeter",
"Definition:Strictly Positive/Integ... |
proofwiki-12242 | Square Modulo 5/Corollary | When written in conventional base $10$ notation, no square number ends in one of $2, 3, 7, 8$. | The absence of $2$ and $3$ from the digit that can end a square follows directly from Square Modulo 5.
As $7 \equiv 2 \pmod 5$ and $8 \equiv 3 \pmod 5$, the result for $7$ and $8$ follows directly.
{{qed}} | When written in conventional [[Definition:Decimal Notation|base $10$ notation]], no [[Definition:Square Number|square number]] ends in one of $2, 3, 7, 8$. | The absence of $2$ and $3$ from the [[Definition:Digit|digit]] that can end a [[Definition:Square Number|square]] follows directly from [[Square Modulo 5]].
As $7 \equiv 2 \pmod 5$ and $8 \equiv 3 \pmod 5$, the result for $7$ and $8$ follows directly.
{{qed}} | Square Modulo 5/Corollary | https://proofwiki.org/wiki/Square_Modulo_5/Corollary | https://proofwiki.org/wiki/Square_Modulo_5/Corollary | [
"Modulo Arithmetic",
"Square Numbers"
] | [
"Definition:Decimal Notation",
"Definition:Square Number"
] | [
"Definition:Digit",
"Definition:Square Number",
"Square Modulo 5"
] |
proofwiki-12243 | Prime equals Plus or Minus One modulo 6 | Let $p$ be a prime number greater than $3$.
Then $p$ is either of the form:
:$p = 6 n + 1$
or:
:$p = 6 n - 1$
That is:
:$p = \pm 1 \pmod 6$ | To demonstrate that there are prime numbers of either form, note:
:$5 = 6 \times 1 - 1$
:$7 = 6 \times 1 + 1$
The only other possibilities for $p$ are:
:$p = 6 n$, in which case $6 \divides p$ and so $p$ is not prime
:$p = 6 n + 2$, in which case $2 \divides p$ and so $p$ is not prime
:$p = 6 n + 3$, in which case $3 \... | Let $p$ be a [[Definition:Prime Number|prime number]] greater than $3$.
Then $p$ is either of the form:
:$p = 6 n + 1$
or:
:$p = 6 n - 1$
That is:
:$p = \pm 1 \pmod 6$ | To demonstrate that there are [[Definition:Prime Number|prime numbers]] of either form, note:
:$5 = 6 \times 1 - 1$
:$7 = 6 \times 1 + 1$
The only other possibilities for $p$ are:
:$p = 6 n$, in which case $6 \divides p$ and so $p$ is not [[Definition:Prime Number|prime]]
:$p = 6 n + 2$, in which case $2 \divides p$ ... | Prime equals Plus or Minus One modulo 6 | https://proofwiki.org/wiki/Prime_equals_Plus_or_Minus_One_modulo_6 | https://proofwiki.org/wiki/Prime_equals_Plus_or_Minus_One_modulo_6 | [
"Prime Numbers",
"6"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-12244 | Number is Sum of Five Cubes | Let $n \in \Z$ be an integer.
Then $n$ can be expressed as the sum of $5$ cubes (either positive or negative) in an infinite number of ways. | We have for any $m, n \in \Z$:
{{begin-eqn}}
{{eqn | l = \paren {6 m + n}^3
| o = \equiv
| r = n^3
| rr = \pmod 6
| c = Congruence of Powers
}}
{{eqn | o = \equiv
| r = n
| rr = \pmod 6
| c = Euler's Theorem (Number Theory): {{EulerPhiLink|6}} $= 2$
}}
{{end-eqn}}
By definition... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $n$ can be expressed as the [[Definition:Integer Addition|sum]] of $5$ [[Definition:Cube Number|cubes]] (either [[Definition:Positive Integer|positive]] or [[Definition:Negative Integer|negative]]) in an [[Definition:Infinite|infinite]] number of ways. | We have for any $m, n \in \Z$:
{{begin-eqn}}
{{eqn | l = \paren {6 m + n}^3
| o = \equiv
| r = n^3
| rr = \pmod 6
| c = [[Congruence of Powers]]
}}
{{eqn | o = \equiv
| r = n
| rr = \pmod 6
| c = [[Euler's Theorem (Number Theory)]]: {{EulerPhiLink|6}} $= 2$
}}
{{end-eqn}}
By d... | Number is Sum of Five Cubes | https://proofwiki.org/wiki/Number_is_Sum_of_Five_Cubes | https://proofwiki.org/wiki/Number_is_Sum_of_Five_Cubes | [
"Cube Numbers",
"Sums of Cubes",
"5"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Positive/Integer",
"Definition:Negative/Integer",
"Definition:Infinite"
] | [
"Congruence of Powers",
"Euler's Theorem (Number Theory)",
"Definition:Modulo Arithmetic",
"Binomial Theorem/Examples/Cube of Sum",
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Infinite Set",
"Definition:Infinite Set"
] |
proofwiki-12245 | Lamé's Theorem | Let $a, b \in \Z_{>0}$ be (strictly) positive integers.
Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation.
Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$.
Then it will take fewer than $5 \times \min \set {c, d}$ integer divisions to find $\gcd... | === Lemma===
{{:Lamé's Theorem/Lemma}}{{qed|lemma}}
{{WLOG}} suppose $a \ge b$.
Then $\min \set {c, d}$ is the number of digits in $b$.
By Number of Digits in Number, we have:
:$\min \set {c, d} = \floor {\log b} + 1$
{{AimForCont}} it takes at least $5 \paren {\floor {\log b} + 1}$ cycles around the Euclidean Algorit... | Let $a, b \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $c$ and $d$ be the number of [[Definition:Digit|digits]] in $a$ and $b$ respectively when expressed in [[Definition:Decimal Notation|decimal notation]].
Let the [[Euclidean Algorithm]] be employed to find the [[Defini... | === [[Lamé's Theorem/Lemma|Lemma]]===
{{:Lamé's Theorem/Lemma}}{{qed|lemma}}
{{WLOG}} suppose $a \ge b$.
Then $\min \set {c, d}$ is the number of [[Definition:Digit|digits]] in $b$.
By [[Number of Digits in Number]], we have:
:$\min \set {c, d} = \floor {\log b} + 1$
{{AimForCont}} it takes at least $5 \paren {\f... | Lamé's Theorem | https://proofwiki.org/wiki/Lamé's_Theorem | https://proofwiki.org/wiki/Lamé's_Theorem | [
"Euclidean Algorithm",
"Lamé's Theorem"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Digit",
"Definition:Decimal Notation",
"Euclidean Algorithm",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Division"
] | [
"Lamé's Theorem/Lemma",
"Definition:Digit",
"Number of Digits in Number",
"Euclidean Algorithm",
"Lamé's Theorem/Lemma",
"Fibonacci Number greater than Golden Section to Power less Two",
"Definition:Contradiction",
"Logarithm of Power/General Logarithm",
"Definition:Contradiction",
"Proof by Contr... |
proofwiki-12246 | Volume of Unit Hypersphere | The volume of the unit sphere in $n$-dimensional space increases as $n$ goes up to $5$, but decreases thereafter. | {{ProofWanted|Needs a considerable amount of background work to be completed first.}} | The [[Definition:Volume|volume]] of the [[Definition:Unit Sphere (Topology)|unit sphere]] in [[Definition:Dimension (Geometry)|$n$-dimensional space]] increases as $n$ goes up to $5$, but decreases thereafter. | {{ProofWanted|Needs a considerable amount of background work to be completed first.}} | Volume of Unit Hypersphere | https://proofwiki.org/wiki/Volume_of_Unit_Hypersphere | https://proofwiki.org/wiki/Volume_of_Unit_Hypersphere | [
"Geometry",
"Spheres"
] | [
"Definition:Volume",
"Definition:Unit Sphere/Topology",
"Definition:Dimension (Geometry)"
] | [] |
proofwiki-12247 | Finite Subset Bounds Element of Finite Infima Set and Upper Closure | Let $L = \struct {S, \wedge, \preceq}$ be meet semilattice.
Let $F$ be filter in $L$.
Let $X$ be non empty finite subset of $S$.
Let $x \in S$ such that
:$x \in \paren {\map {\operatorname{fininfs} } {F \cup X} }^\succeq$
where
:$\operatorname{fininfs}$ denotes the finite infima set
:$X^\succeq$ denotes the upper closu... | By definition of upper closure of subset:
:$\exists u \in \map {\operatorname{fininfs} } {F \cup X}: u \preceq x$
By definition of finite infima set:
:there exists finite subset $Y$ of $F \cup X$:
:$Y$ admits an infimum and $u = \inf Y$
We will prove that
:$Y \setminus X \subseteq F$
Let $a \in Y \setminus X$.
By defin... | Let $L = \struct {S, \wedge, \preceq}$ be [[Definition:Meet Semilattice|meet semilattice]].
Let $F$ be [[Definition:Filter|filter]] in $L$.
Let $X$ be [[Definition:Non-Empty Set|non empty]] [[Definition:Finite Subset|finite subset]] of $S$.
Let $x \in S$ such that
:$x \in \paren {\map {\operatorname{fininfs} } {F \c... | By definition of [[Definition:Upper Closure of Subset|upper closure of subset]]:
:$\exists u \in \map {\operatorname{fininfs} } {F \cup X}: u \preceq x$
By definition of [[Definition:Finite Infima Set|finite infima set]]:
:there exists [[Definition:Finite Subset|finite subset]] $Y$ of $F \cup X$:
:$Y$ admits an [[Defi... | Finite Subset Bounds Element of Finite Infima Set and Upper Closure | https://proofwiki.org/wiki/Finite_Subset_Bounds_Element_of_Finite_Infima_Set_and_Upper_Closure | https://proofwiki.org/wiki/Finite_Subset_Bounds_Element_of_Finite_Infima_Set_and_Upper_Closure | [
"Join and Meet Semilattices",
"Upper Closures"
] | [
"Definition:Meet Semilattice",
"Definition:Filter",
"Definition:Non-Empty Set",
"Definition:Finite Subset",
"Definition:Finite Infima Set",
"Definition:Upper Closure/Set"
] | [
"Definition:Upper Closure/Set",
"Definition:Finite Infima Set",
"Definition:Finite Subset",
"Definition:Infimum of Set",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Set Union",
"Definition:Non-Empty Set",
"Meet Precedes Operands",
"Set Difference with Superset is Empty Set",
"E... |
proofwiki-12248 | Length of Reciprocal of Product of Powers of 2 and 5 | Let $n \in \Z$ be an integer.
Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits.
Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$. | Since $\dfrac 1 n$ terminates after $m$ digits:
:$\dfrac {10^m} n$ is an integer
:$\dfrac {10^{m - 1}} n$ is not an integer
From the first condition, we have $n = 2^p 5^q$ for some positive integers $p, q \le m$.
This gives $m \ge \max \set {p, q}$.
From the second condition, we cannot have both $p, q \le m - 1$.
There... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $\dfrac 1 n$, when expressed as a [[Definition:Decimal Expansion|decimal expansion]], [[Definition:Termination of Basis Expansion|terminate]] after $m$ digits.
Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$. | Since $\dfrac 1 n$ [[Definition:Termination of Basis Expansion|terminates]] after $m$ digits:
:$\dfrac {10^m} n$ is an [[Definition:Integer|integer]]
:$\dfrac {10^{m - 1}} n$ is not an [[Definition:Integer|integer]]
From the first condition, we have $n = 2^p 5^q$ for some [[Definition:Positive Integer|positive intege... | Length of Reciprocal of Product of Powers of 2 and 5 | https://proofwiki.org/wiki/Length_of_Reciprocal_of_Product_of_Powers_of_2_and_5 | https://proofwiki.org/wiki/Length_of_Reciprocal_of_Product_of_Powers_of_2_and_5 | [
"Number Theory",
"Reciprocals"
] | [
"Definition:Integer",
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Termination"
] | [
"Definition:Basis Expansion/Termination",
"Definition:Integer",
"Definition:Integer",
"Definition:Positive/Integer"
] |
proofwiki-12249 | Structure of Recurring Decimal | Let $\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits with no nonperiodic part.
Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $q$ digits.
Then $\dfrac 1 {m n}$ has a nonperiodic part of $q$ digits, and a recurring part of $p$ digits. | Let $b \in \N_{>1}$ be the base we are working on.
Note that $b^p \times \dfrac 1 m$ is the result of shifting the decimal point of $\dfrac 1 m$ by $p$ digits.
Hence $b^p \times \dfrac 1 m - \dfrac 1 m$ is an integer, and $\paren {b^i - 1} \dfrac 1 m$ is not an integer for integers $0 < i < p$.
Therefore $m \divides b^... | Let $\dfrac 1 m$, when expressed as a [[Definition:Decimal Expansion|decimal expansion]], [[Definition:Recurrence of Basis Expansion|recur]] with a [[Definition:Period of Recurrence|period]] of $p$ [[Definition:Digit|digits]] with no [[Definition:Non-Recurring Part of Recurring Basis Expansion|nonperiodic part]].
Let ... | Let $b \in \N_{>1}$ be the [[Definition:Number Base|base]] we are working on.
Note that $b^p \times \dfrac 1 m$ is the result of shifting the decimal point of $\dfrac 1 m$ by $p$ [[Definition:Digit|digits]].
Hence $b^p \times \dfrac 1 m - \dfrac 1 m$ is an [[Definition:Integer|integer]], and $\paren {b^i - 1} \dfrac ... | Structure of Recurring Decimal | https://proofwiki.org/wiki/Structure_of_Recurring_Decimal | https://proofwiki.org/wiki/Structure_of_Recurring_Decimal | [
"Number Theory"
] | [
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Recurrence",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Non-Recurring Part",
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Termination",
"Definition:Digit",
"Defi... | [
"Definition:Number Base",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Division Theorem",
"Definition:Basis Expansion/Recurrence/Non-Recurring Part",
"Definition:Digi... |
proofwiki-12250 | Characteristics of Pentatope | A pentatope has $5$ cells, $10$ faces, $10$ edges and $5$ vertices. | {{ProofWanted|... also worth mentioning that an astute reader would correlate this result with the similar characteristics of a tetrahedron and triangle, and wonder whether Pascal's triangle comes into this anywhere.}} | A [[Definition:Pentatope|pentatope]] has $5$ [[Definition:Cell of Polytope|cells]], $10$ [[Definition:Face of Polytope|faces]], $10$ [[Definition:Edge of Polytope|edges]] and $5$ [[Definition:Vertex of Polytope|vertices]]. | {{ProofWanted|... also worth mentioning that an astute reader would correlate this result with the similar characteristics of a tetrahedron and triangle, and wonder whether Pascal's triangle comes into this anywhere.}} | Characteristics of Pentatope | https://proofwiki.org/wiki/Characteristics_of_Pentatope | https://proofwiki.org/wiki/Characteristics_of_Pentatope | [
"Pentatopes"
] | [
"Definition:Pentatope",
"Definition:Cell of Polytope",
"Definition:Face of Polytope",
"Definition:Edge of Polytope",
"Definition:Vertex of Polytope"
] | [] |
proofwiki-12251 | Pentatope is Self-Dual | A pentatope is self-dual. | {{ProofWanted|Plenty of background work on hyperdimensional geometry to be done before we can even start.}} | A [[Definition:Pentatope|pentatope]] is [[Definition:Self-Dual|self-dual]]. | {{ProofWanted|Plenty of background work on hyperdimensional geometry to be done before we can even start.}} | Pentatope is Self-Dual | https://proofwiki.org/wiki/Pentatope_is_Self-Dual | https://proofwiki.org/wiki/Pentatope_is_Self-Dual | [
"Pentatopes"
] | [
"Definition:Pentatope",
"Definition:Self-Dual"
] | [] |
proofwiki-12252 | Exponential is of Exponential Order Real Part of Index | Let $\map f t = e^{\psi t}$ be the complex exponential function, where $t \in \R, \psi \in \C$.
Let $a = \map \Re \psi$.
Then $e^{\psi t}$ is of exponential order $a$. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {e^{\psi t} }
| r = e^{a t}
| c = Modulus of Exponential is Exponential of Real Part
}}
{{eqn | o = <
| r = 2 e^{a t}
| c = Exponential of Real Number is Strictly Positive
}}
{{end-eqn}}
The result follows from the definition of exponen... | Let $\map f t = e^{\psi t}$ be the [[Definition:Complex Exponential Function|complex exponential function]], where $t \in \R, \psi \in \C$.
Let $a = \map \Re \psi$.
Then $e^{\psi t}$ is of [[Definition:Exponential Order to Real Index|exponential order $a$]]. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {e^{\psi t} }
| r = e^{a t}
| c = [[Modulus of Exponential is Exponential of Real Part]]
}}
{{eqn | o = <
| r = 2 e^{a t}
| c = [[Exponential of Real Number is Strictly Positive]]
}}
{{end-eqn}}
The result follows from the definition o... | Exponential is of Exponential Order Real Part of Index | https://proofwiki.org/wiki/Exponential_is_of_Exponential_Order_Real_Part_of_Index | https://proofwiki.org/wiki/Exponential_is_of_Exponential_Order_Real_Part_of_Index | [
"Exponential Order"
] | [
"Definition:Exponential Function/Complex",
"Definition:Exponential Order/Real Index"
] | [
"Modulus of Exponential is Exponential of Real Part",
"Exponential of Real Number is Strictly Positive",
"Definition:Exponential Order/Real Index",
"Category:Exponential Order"
] |
proofwiki-12253 | Cosine is of Exponential Order Zero | Let $\cos t$ be the cosine of $t$, where $t \in \R$.
Then $\cos t$ is of exponential order $0$. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {\cos t}
| o = \le
| r = 1
| c = Real Cosine Function is Bounded
}}
{{eqn | ll= \leadsto
| l = \size {\cos t}
| o = <
| r = 2
}}
{{eqn | r = 2 e^{0 t}
| c = Exponential of Zero
}}
{{end-eqn}}
The result follows from th... | Let $\cos t$ be the [[Definition:Real Cosine Function|cosine of $t$]], where $t \in \R$.
Then $\cos t$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size {\cos t}
| o = \le
| r = 1
| c = [[Real Cosine Function is Bounded]]
}}
{{eqn | ll= \leadsto
| l = \size {\cos t}
| o = <
| r = 2
}}
{{eqn | r = 2 e^{0 t}
| c = [[Exponential of Zero]]
}}
{{end-eqn}}
The result follow... | Cosine is of Exponential Order Zero/Proof 1 | https://proofwiki.org/wiki/Cosine_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Cosine_is_of_Exponential_Order_Zero/Proof_1 | [
"Exponential Order",
"Cosine Function",
"Cosine is of Exponential Order Zero"
] | [
"Definition:Cosine/Real Function",
"Definition:Exponential Order/Real Index"
] | [
"Real Cosine Function is Bounded",
"Exponential of Zero",
"Definition:Exponential Order/Real Index"
] |
proofwiki-12254 | Cosine is of Exponential Order Zero | Let $\cos t$ be the cosine of $t$, where $t \in \R$.
Then $\cos t$ is of exponential order $0$. | The result follows from Real Cosine Function is Bounded and Bounded Function is of Exponential Order Zero.
{{qed}} | Let $\cos t$ be the [[Definition:Real Cosine Function|cosine of $t$]], where $t \in \R$.
Then $\cos t$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | The result follows from [[Real Cosine Function is Bounded]] and [[Bounded Function is of Exponential Order Zero]].
{{qed}} | Cosine is of Exponential Order Zero/Proof 2 | https://proofwiki.org/wiki/Cosine_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Cosine_is_of_Exponential_Order_Zero/Proof_2 | [
"Exponential Order",
"Cosine Function",
"Cosine is of Exponential Order Zero"
] | [
"Definition:Cosine/Real Function",
"Definition:Exponential Order/Real Index"
] | [
"Real Cosine Function is Bounded",
"Bounded Function is of Exponential Order Zero"
] |
proofwiki-12255 | Sine is of Exponential Order Zero | Let $\sin t$ be the sine of $t$, where $t \in \R$.
Then $\sin t$ is of exponential order $0$. | {{begin-eqn}}
{{eqn | l = \size {\sin t}
| o = \le
| r = 1
| c = Real Sine Function is Bounded
}}
{{eqn | ll =\leadsto
| l = \size {\sin t}
| o = <
| r = 2
}}
{{eqn | r = 2 e^{0 t}
| c = Exponential of Zero
}}
{{end-eqn}}
{{qed}} | Let $\sin t$ be the [[Definition:Sine|sine of $t$]], where $t \in \R$.
Then $\sin t$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | {{begin-eqn}}
{{eqn | l = \size {\sin t}
| o = \le
| r = 1
| c = [[Real Sine Function is Bounded]]
}}
{{eqn | ll =\leadsto
| l = \size {\sin t}
| o = <
| r = 2
}}
{{eqn | r = 2 e^{0 t}
| c = [[Exponential of Zero]]
}}
{{end-eqn}}
{{qed}} | Sine is of Exponential Order Zero/Proof 1 | https://proofwiki.org/wiki/Sine_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Sine_is_of_Exponential_Order_Zero/Proof_1 | [
"Exponential Order",
"Sine Function",
"Sine is of Exponential Order Zero"
] | [
"Definition:Sine",
"Definition:Exponential Order/Real Index"
] | [
"Real Sine Function is Bounded",
"Exponential of Zero"
] |
proofwiki-12256 | Scalar Multiple of Function of Exponential Order | Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$.
Let $\lambda$ be a complex constant.
Suppose $f$ is of exponential order $a$.
Then $\lambda f$ is also of exponential order $a$. | If $\lambda = 0$, the theorem holds trivially.
Let $\lambda \ne 0$.
{{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K e^{a t}
| c = {{Defof|Exponential Order to Real Index}}
}}
{{eqn | ll= \leadsto
| l = \size \lambda \size {\map f t}
| o = <
| r = \size \lambda K e^{a t}
}}... | Let $f: \R \to \F$ be a [[Definition:Function|function]], where $\F \in \set {\R, \C}$.
Let $\lambda$ be a [[Definition:Complex Number|complex constant]].
Suppose $f$ is of [[Definition:Exponential Order to Real Index|exponential order $a$]].
Then $\lambda f$ is also of [[Definition:Exponential Order to Real Index|... | If $\lambda = 0$, the theorem holds trivially.
Let $\lambda \ne 0$.
{{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K e^{a t}
| c = {{Defof|Exponential Order to Real Index}}
}}
{{eqn | ll= \leadsto
| l = \size \lambda \size {\map f t}
| o = <
| r = \size \lambda K e^{a t}
... | Scalar Multiple of Function of Exponential Order | https://proofwiki.org/wiki/Scalar_Multiple_of_Function_of_Exponential_Order | https://proofwiki.org/wiki/Scalar_Multiple_of_Function_of_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Complex Number",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index"
] | [
"Complex Modulus of Product of Complex Numbers",
"Category:Exponential Order"
] |
proofwiki-12257 | Function of Exponential Order of Scalar Multiple | Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$.
Let $\lambda$ be a real constant.
Let $\map f t$ be of exponential order $a$.
Then the function defined by $t \mapsto \map f {\lambda t}$ is of exponential order $a\lambda$. | {{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K e^{a t}
| c = {{Defof|Exponential Order to Real Index}}
}}
{{eqn | ll= \leadsto
| l = \size {\map f {\lambda t} }
| o = <
| r = K e^{a \lambda t}
| c = replacing $t$ with $\lambda t$
}}
{{end-eqn}}
The result follows by... | Let $f: \R \to \F$ be a [[Definition:Function|function]], where $\F \in \set {\R, \C}$.
Let $\lambda$ be a [[Definition:Real Number|real constant]].
Let $\map f t$ be of [[Definition:Exponential Order to Real Index|exponential order $a$]].
Then the function defined by $t \mapsto \map f {\lambda t}$ is of [[Definiti... | {{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K e^{a t}
| c = {{Defof|Exponential Order to Real Index}}
}}
{{eqn | ll= \leadsto
| l = \size {\map f {\lambda t} }
| o = <
| r = K e^{a \lambda t}
| c = replacing $t$ with $\lambda t$
}}
{{end-eqn}}
The result follows b... | Function of Exponential Order of Scalar Multiple | https://proofwiki.org/wiki/Function_of_Exponential_Order_of_Scalar_Multiple | https://proofwiki.org/wiki/Function_of_Exponential_Order_of_Scalar_Multiple | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Real Number",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Exponential Order/Real Index",
"Category:Exponential Order"
] |
proofwiki-12258 | Identity is of Exponential Order Epsilon | Let $I_\R: t \mapsto t$ be the identity mapping on $\R_{\ge 0}$.
Then $I_\R$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | {{begin-eqn}}
{{eqn | l = e^{\epsilon t}
| o = \ge
| r = 1 + \epsilon t
| c = Exponential of $t$ not less than $1 + t$
}}
{{eqn | o = >
| r = \epsilon t
}}
{{eqn | ll= \leadsto
| l = K e^{\epsilon t}
| o = >
| r = t
| c = $K = \dfrac 1 \epsilon$
}}
{{end-eqn}}
{{qed}}
Cat... | Let $I_\R: t \mapsto t$ be the [[Definition:Identity Mapping|identity mapping]] on $\R_{\ge 0}$.
Then $I_\R$ is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | {{begin-eqn}}
{{eqn | l = e^{\epsilon t}
| o = \ge
| r = 1 + \epsilon t
| c = [[Exponential of x not less than 1+x|Exponential of $t$ not less than $1 + t$]]
}}
{{eqn | o = >
| r = \epsilon t
}}
{{eqn | ll= \leadsto
| l = K e^{\epsilon t}
| o = >
| r = t
| c = $K = \dfrac... | Identity is of Exponential Order Epsilon | https://proofwiki.org/wiki/Identity_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Identity_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Identity Mapping",
"Definition:Exponential Order/Real Index"
] | [
"Exponential of x not less than 1+x",
"Category:Exponential Order"
] |
proofwiki-12259 | Product of Functions of Exponential Order | Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$.
Let $f$ be of exponential order $a$ and $g$ be of exponential order $b$.
Then $f g: t \mapsto \map f t \map g t$ is of exponential order $a+b$. | Let $t$ be sufficiently large so that both $f$ and $g$ are of exponential order on some shared unbounded closed interval.
By the definition of exponential order:
{{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K_1 e^{a t}
}}
{{eqn | l = \size {\map g t}
| o = <
| r = K_2 e^{b t}
}}
{{eq... | Let $f, g: \R \to \F$ be [[Definition:Function|functions]], where $\F \in \set {\R, \C}$.
Let $f$ be of [[Definition:Exponential Order to Real Index|exponential order $a$]] and $g$ be of [[Definition:Exponential Order to Real Index|exponential order $b$]].
Then $f g: t \mapsto \map f t \map g t$ is of [[Definition:E... | Let $t$ be [[Definition:Sufficiently Large|sufficiently large]] so that both $f$ and $g$ are of [[Definition:Exponential Order|exponential order]] on some shared [[Definition:Unbounded Closed Real Interval|unbounded closed interval]].
By the definition of [[Definition:Exponential Order|exponential order]]:
{{begin-eq... | Product of Functions of Exponential Order | https://proofwiki.org/wiki/Product_of_Functions_of_Exponential_Order | https://proofwiki.org/wiki/Product_of_Functions_of_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Sufficiently Large",
"Definition:Exponential Order",
"Definition:Real Interval/Unbounded Closed",
"Definition:Exponential Order",
"Positive Real Number Inequalities can be Multiplied",
"Complex Modulus of Product of Complex Numbers",
"Exponential of Sum/Real Numbers",
"Category:Exponential... |
proofwiki-12260 | Sum of Functions of Exponential Order | Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$.
Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$.
Then $f + g: t \mapsto \map f t + \map g t$ is of exponential order $\max \set {a, b}$. | Let $t$ be sufficiently large so that both $f$ and $g$ are of exponential order on some shared unbounded closed interval.
By the definition of exponential order:
{{begin-eqn}}
{{eqn | l = \size {\map f t}
| o = <
| r = K_1 e^{a t}
}}
{{eqn | l = \size {\map g t}
| o = <
| r = K_2 e^{b t}
}}
{{eq... | Let $f, g: \R \to \F$ be [[Definition:Function|functions]], where $\F \in \set {\R, \C}$.
Suppose $f$ is of [[Definition:Exponential Order to Real Index|exponential order $a$]] and $g$ is of [[Definition:Exponential Order to Real Index|exponential order $b$]].
Then $f + g: t \mapsto \map f t + \map g t$ is of [[Defi... | Let $t$ be [[Definition:Sufficiently Large|sufficiently large]] so that both $f$ and $g$ are of [[Definition:Exponential Order|exponential order]] on some shared [[Definition:Unbounded Closed Real Interval|unbounded closed interval]].
By the definition of [[Definition:Exponential Order|exponential order]]:
{{begin-eq... | Sum of Functions of Exponential Order | https://proofwiki.org/wiki/Sum_of_Functions_of_Exponential_Order | https://proofwiki.org/wiki/Sum_of_Functions_of_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Sufficiently Large",
"Definition:Exponential Order",
"Definition:Real Interval/Unbounded Closed",
"Definition:Exponential Order",
"Real Number Inequalities can be Added",
"Triangle Inequality/Real Numbers",
"Exponential is Strictly Increasing",
"Category:Exponential Order"
] |
proofwiki-12261 | Linear Combination of Functions of Exponential Order | Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$.
Let $\lambda, \mu$ be complex numbers.
Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$.
Then $\map {\paren {\lambda f + \mu g} } t = \lambda \, \map f t + \mu \, \map g t$ is of exponential order $\max \set {a, b}$. | Follows from:
* Scalar Multiple of Function of Exponential Order
* Sum of Functions of Exponential Order
{{qed}}
Category:Exponential Order
l0rjg4v33skvpaty328lncl3s7n5zut | Let $f, g: \R \to \F$ be [[Definition:Function|functions]], where $\F \in \set {\R, \C}$.
Let $\lambda, \mu$ be [[Definition:Complex Number|complex numbers]].
Suppose $f$ is of [[Definition:Exponential Order to Real Index|exponential order $a$]] and $g$ is of [[Definition:Exponential Order to Real Index|exponential ... | Follows from:
* [[Scalar Multiple of Function of Exponential Order]]
* [[Sum of Functions of Exponential Order]]
{{qed}}
[[Category:Exponential Order]]
l0rjg4v33skvpaty328lncl3s7n5zut | Linear Combination of Functions of Exponential Order | https://proofwiki.org/wiki/Linear_Combination_of_Functions_of_Exponential_Order | https://proofwiki.org/wiki/Linear_Combination_of_Functions_of_Exponential_Order | [
"Exponential Order"
] | [
"Definition:Function",
"Definition:Complex Number",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index",
"Definition:Exponential Order/Real Index"
] | [
"Scalar Multiple of Function of Exponential Order",
"Sum of Functions of Exponential Order",
"Category:Exponential Order"
] |
proofwiki-12262 | Constant Function is of Exponential Order Zero | Let $f_C: \R \to \GF: t \mapsto C$ be a constant function, where $\GF \in \set {\R, \C}$.
Then $f_C$ is of exponential order $0$. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size C
| o = <
| r = \size C + 1
}}
{{eqn | r = \paren {\size C + 1} e^{0 t}
| c = Exponential of Zero
}}
{{end-eqn}}
The result follows from the definition of exponential order, with $M = 1$, $K = \size C + 1$, and $a = 0$.
{{qed}}
Category:Expon... | Let $f_C: \R \to \GF: t \mapsto C$ be a [[Definition:Constant Function|constant function]], where $\GF \in \set {\R, \C}$.
Then $f_C$ is of [[Definition:Exponential Order to Real Index|exponential order $0$]]. | {{begin-eqn}}
{{eqn | q = \forall t \ge 1
| l = \size C
| o = <
| r = \size C + 1
}}
{{eqn | r = \paren {\size C + 1} e^{0 t}
| c = [[Exponential of Zero]]
}}
{{end-eqn}}
The result follows from the definition of [[Definition:Exponential Order to Real Index|exponential order]], with $M = 1$, $K... | Constant Function is of Exponential Order Zero | https://proofwiki.org/wiki/Constant_Function_is_of_Exponential_Order_Zero | https://proofwiki.org/wiki/Constant_Function_is_of_Exponential_Order_Zero | [
"Exponential Order"
] | [
"Definition:Constant Mapping",
"Definition:Exponential Order/Real Index"
] | [
"Exponential of Zero",
"Definition:Exponential Order/Real Index",
"Category:Exponential Order"
] |
proofwiki-12263 | Polynomial is of Exponential Order Epsilon | Let $P: \R \to \mathbb F$ be a polynomial, where $\mathbb F \in \set {\R, \C}$.
Then $P$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | If $P = 0$, the theorem holds trivially.
Let $P_n$ be a polynomial of degree $n$, where $n \ge 0$.
The proof proceeds by induction on $n$, where $n$ is the degree of the polynomial. | Let $P: \R \to \mathbb F$ be a [[Definition:Polynomial (Analysis)|polynomial]], where $\mathbb F \in \set {\R, \C}$.
Then $P$ is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | If $P = 0$, the theorem holds trivially.
Let $P_n$ be a [[Definition:Polynomial (Analysis)|polynomial]] of [[Definition:Degree of Polynomial over Field|degree $n$]], where $n \ge 0$.
The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$, where $n$ is the [[Definition:Degree of Polynomial|deg... | Polynomial is of Exponential Order Epsilon | https://proofwiki.org/wiki/Polynomial_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Polynomial_is_of_Exponential_Order_Epsilon | [
"Exponential Order",
"Proofs by Induction"
] | [
"Definition:Polynomial",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Polynomial",
"Definition:Degree of Polynomial/Field",
"Principle of Mathematical Induction",
"Definition:Degree of Polynomial",
"Definition:Degree of Polynomial",
"Definition:Degree of Polynomial/Field",
"Definition:Polynomial",
"Principle of Mathematical Induction"
] |
proofwiki-12264 | Fibonacci Number by Power of 2 | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 2^{n - 1} F_n
| r = \sum_k 5^k \dbinom n {2 k + 1}
| c =
}}
{{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots
| c =
}}
{{end-eqn}}
where:
:$F_n$ denotes the $n$th Fibonacci number
:$\dbinom n {2 k + 1} \ $ denotes a bino... | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds 2^{n - 1} F_n = \sum_k 5^k \dbinom n {2 k + 1}$
First note the bounds of the summation.
By definition, $\dbinom n k = 0$ where $k < 0$ or $k > n$.
Thus in all cases in the following, terms outside the range $0 \le k \l... | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 2^{n - 1} F_n
| r = \sum_k 5^k \dbinom n {2 k + 1}
| c =
}}
{{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots
| c =
}}
{{end-eqn}}
where:
:$F_n$ denotes the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]]
:$\dbin... | 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]]:
:$\ds 2^{n - 1} F_n = \sum_k 5^k \dbinom n {2 k + 1}$
First note the bounds of the [[Definition:Summation|summation]].
By definition, $\dbinom n k = 0$ w... | Fibonacci Number by Power of 2/Proof 1 | https://proofwiki.org/wiki/Fibonacci_Number_by_Power_of_2 | https://proofwiki.org/wiki/Fibonacci_Number_by_Power_of_2/Proof_1 | [
"Fibonacci Numbers",
"Binomial Coefficients",
"Fibonacci Number by Power of 2"
] | [
"Definition:Fibonacci Number",
"Definition:Binomial Coefficient"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Summation",
"Zero Choose n",
"One Choose n",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Fibonacci Number by Power of 2",
"Principle of Mathematical Induction"
] |
proofwiki-12265 | Fibonacci Number by Power of 2 | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 2^{n - 1} F_n
| r = \sum_k 5^k \dbinom n {2 k + 1}
| c =
}}
{{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots
| c =
}}
{{end-eqn}}
where:
:$F_n$ denotes the $n$th Fibonacci number
:$\dbinom n {2 k + 1} \ $ denotes a bino... | {{begin-eqn}}
{{eqn | l = 2^{n - 1} F_n
| r = \dfrac {2^n} {2 \sqrt 5} \paren {\phi^n - \hat \phi^n}
| c = Euler-Binet Formula
}}
{{eqn | r = \dfrac {\paren {1 + \sqrt 5}^n - \paren {1 - \sqrt 5}^n} {2 \sqrt 5}
| c = {{Defof|Golden Mean|index = 2}}
}}
{{eqn | r = \dfrac 1 {2 \sqrt 5} \sum_{j \mathop =... | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 2^{n - 1} F_n
| r = \sum_k 5^k \dbinom n {2 k + 1}
| c =
}}
{{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots
| c =
}}
{{end-eqn}}
where:
:$F_n$ denotes the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]]
:$\dbin... | {{begin-eqn}}
{{eqn | l = 2^{n - 1} F_n
| r = \dfrac {2^n} {2 \sqrt 5} \paren {\phi^n - \hat \phi^n}
| c = [[Euler-Binet Formula]]
}}
{{eqn | r = \dfrac {\paren {1 + \sqrt 5}^n - \paren {1 - \sqrt 5}^n} {2 \sqrt 5}
| c = {{Defof|Golden Mean|index = 2}}
}}
{{eqn | r = \dfrac 1 {2 \sqrt 5} \sum_{j \math... | Fibonacci Number by Power of 2/Proof 2 | https://proofwiki.org/wiki/Fibonacci_Number_by_Power_of_2 | https://proofwiki.org/wiki/Fibonacci_Number_by_Power_of_2/Proof_2 | [
"Fibonacci Numbers",
"Binomial Coefficients",
"Fibonacci Number by Power of 2"
] | [
"Definition:Fibonacci Number",
"Definition:Binomial Coefficient"
] | [
"Euler-Binet Formula",
"Binomial Theorem"
] |
proofwiki-12266 | Ratio of 2016 to Aliquot Sum | $2016$ has the property that its ratio to its aliquot sum is $4 : 9$. | The aliquot sum of an integer $n$ is the integer sum of the aliquot parts of $n$.
That is, the aliquot sum of $2016$ is the divisor sum of $2016$ minus $2016$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {2016} - 2016
| r = 6552 - 2016
| c = {{DSFLink|2016}}
}}
{{eqn | r = 4536
| c =
}}
{{eqn | ... | $2016$ has the property that its [[Definition:Ratio|ratio]] to its [[Definition:Aliquot Sum|aliquot sum]] is $4 : 9$. | The [[Definition:Aliquot Sum|aliquot sum]] of an [[Definition:Integer|integer]] $n$ is the [[Definition:Integer Addition|integer sum]] of the [[Definition:Aliquot Part|aliquot parts]] of $n$.
That is, the [[Definition:Aliquot Sum|aliquot sum]] of $2016$ is the [[Definition:Divisor Sum Function|divisor sum]] of $2016$... | Ratio of 2016 to Aliquot Sum | https://proofwiki.org/wiki/Ratio_of_2016_to_Aliquot_Sum | https://proofwiki.org/wiki/Ratio_of_2016_to_Aliquot_Sum | [
"2016",
"Aliquot Sums"
] | [
"Definition:Ratio",
"Definition:Aliquot Sum"
] | [
"Definition:Aliquot Sum",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Aliquot Sum",
"Definition:Divisor Sum Function"
] |
proofwiki-12267 | Approximation to Golden Rectangle using Fibonacci Squares | An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner:
:800px
It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing q... | Let the last two squares to be added have side lengths of $F_{n - 1}$ and $F_n$.
Then from the method of construction, the sides of the rectangle generated will be $F_n$ and $F_{n + 1}$.
From Continued Fraction Expansion of Golden Mean it follows that the limit of the ratio of the side lengths of the rectangle, as $n$ ... | An approximation to a [[Definition:Golden Rectangle|golden rectangle]] can be obtained by placing adjacent to one another [[Definition:Square (Geometry)|squares]] with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] corresponding to consecutive [[Definition:Fibonacci Numbers|Fibonacci numbers]... | Let the last two [[Definition:Square (Geometry)|squares]] to be added have [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] of $F_{n - 1}$ and $F_n$.
Then from the method of construction, the [[Definition:Side of Polygon|sides]] of the [[Definition:Rectangle|rectangle]] generated will be $F_n$... | Approximation to Golden Rectangle using Fibonacci Squares/Proof 1 | https://proofwiki.org/wiki/Approximation_to_Golden_Rectangle_using_Fibonacci_Squares | https://proofwiki.org/wiki/Approximation_to_Golden_Rectangle_using_Fibonacci_Squares/Proof_1 | [
"Fibonacci Numbers",
"Golden Mean",
"Approximation to Golden Rectangle using Fibonacci Squares"
] | [
"Definition:Golden Rectangle",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Fibonacci Number",
"File:FibonacciRectangle.png",
"Sequence of Golden Rectangles",
"Definition:Logarithmic Spiral",
"Definition:Circle"
] | [
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Quadrilateral/Rectangle",
"Continued Fraction Expansion of Golden Mean",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Ratio",
"Definition:Polygon/Side"... |
proofwiki-12268 | Approximation to Golden Rectangle using Fibonacci Squares | An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner:
:800px
It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing q... | From Sum of Sequence of Squares of Fibonacci Numbers:
:$\forall n \ge 1: \ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$
Hence the result.
{{qed}} | An approximation to a [[Definition:Golden Rectangle|golden rectangle]] can be obtained by placing adjacent to one another [[Definition:Square (Geometry)|squares]] with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] corresponding to consecutive [[Definition:Fibonacci Numbers|Fibonacci numbers]... | From [[Sum of Sequence of Squares of Fibonacci Numbers]]:
:$\forall n \ge 1: \ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$
Hence the result.
{{qed}} | Approximation to Golden Rectangle using Fibonacci Squares/Proof 2 | https://proofwiki.org/wiki/Approximation_to_Golden_Rectangle_using_Fibonacci_Squares | https://proofwiki.org/wiki/Approximation_to_Golden_Rectangle_using_Fibonacci_Squares/Proof_2 | [
"Fibonacci Numbers",
"Golden Mean",
"Approximation to Golden Rectangle using Fibonacci Squares"
] | [
"Definition:Golden Rectangle",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Fibonacci Number",
"File:FibonacciRectangle.png",
"Sequence of Golden Rectangles",
"Definition:Logarithmic Spiral",
"Definition:Circle"
] | [
"Sum of Sequence of Squares of Fibonacci Numbers"
] |
proofwiki-12269 | Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers | Let $F_n$ denote the $n$th Fibonacci number.
Then:
:$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$ | From Honsberger's Identity:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
Setting $m = n$:
{{begin-eqn}}
{{eqn | l = F_{2 n}
| r = F_{n - 1} F_n + F_n F_{n + 1}
| c =
}}
{{eqn | r = F_n \paren {F_{n + 1} + F_{n - 1} }
| c =
}}
{{eqn | r = \paren {F_{n + 1} - F_{n - 1} } \par... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then:
:$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$ | From [[Honsberger's Identity]]:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
Setting $m = n$:
{{begin-eqn}}
{{eqn | l = F_{2 n}
| r = F_{n - 1} F_n + F_n F_{n + 1}
| c =
}}
{{eqn | r = F_n \paren {F_{n + 1} + F_{n - 1} }
| c =
}}
{{eqn | r = \paren {F_{n + 1} - F_{n - 1... | Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers | https://proofwiki.org/wiki/Fibonacci_Number_of_Index_2n_as_Sum_of_Squares_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Fibonacci_Number_of_Index_2n_as_Sum_of_Squares_of_Fibonacci_Numbers | [
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Honsberger's Identity",
"Difference of Two Squares"
] |
proofwiki-12270 | Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers | Let $F_n$ denote the $n$th Fibonacci number.
Then:
:$F_{3 n} = {F_{n + 1} }^3 + {F_n}^3 - {F_{n - 1} }^3$ | From Honsberger's Identity:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
Setting $m = 2 n$:
{{begin-eqn}}
{{eqn | l = F_{3 n}
| r = F_{2 n + n}
| c =
}}
{{eqn | r = F_{2 n - 1} F_n + F_{2 n} F_{n + 1}
| c = Honsberger's Identity
}}
{{eqn | r = F_{n + \paren{n - 1} } F_n + F_... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then:
:$F_{3 n} = {F_{n + 1} }^3 + {F_n}^3 - {F_{n - 1} }^3$ | From [[Honsberger's Identity]]:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
Setting $m = 2 n$:
{{begin-eqn}}
{{eqn | l = F_{3 n}
| r = F_{2 n + n}
| c =
}}
{{eqn | r = F_{2 n - 1} F_n + F_{2 n} F_{n + 1}
| c = [[Honsberger's Identity]]
}}
{{eqn | r = F_{n + \paren{n - 1... | Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers | https://proofwiki.org/wiki/Fibonacci_Number_of_Index_3n_as_Sum_of_Cubes_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Fibonacci_Number_of_Index_3n_as_Sum_of_Cubes_of_Fibonacci_Numbers | [
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Honsberger's Identity",
"Honsberger's Identity",
"Honsberger's Identity",
"Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers"
] |
proofwiki-12271 | Pythagorean Triangle from Fibonacci Numbers | Take $4$ consecutive Fibonacci numbers:
:$F_n, F_{n + 1}, F_{n + 2}, F_{n + 3}$
Let:
:$a := F_n F_{n + 3}$
:$b := 2 F_{n + 1} F_{n + 2}$
:$c := F_{2 n + 3}$
Then:
:$a^2 + b^2 = c^2$
and:
:$\dfrac {a b} 2 = F_n \times F_{n + 1} \times F_{n + 2} \times F_{n + 3}$
That is, if the legs of a right triangle are the product o... | By definition of Fibonacci numbers:
:$F_n = F_{n + 2} - F_{n + 1}$
and:
:$F_{n + 3} = F_{n + 2} + F_{n + 1}$
Then $a$ can be expressed as:
{{begin-eqn}}
{{eqn | l = a
| r = \paren {F_{n + 2} - F_{n + 1} } \paren {F_{n + 2} + F_{n + 1} }
| c =
}}
{{eqn | r = {F_{n + 2} }^2 - {F_{n + 1} }^2
| c =
}}
{... | Take $4$ consecutive [[Definition:Fibonacci Numbers|Fibonacci numbers]]:
:$F_n, F_{n + 1}, F_{n + 2}, F_{n + 3}$
Let:
:$a := F_n F_{n + 3}$
:$b := 2 F_{n + 1} F_{n + 2}$
:$c := F_{2 n + 3}$
Then:
:$a^2 + b^2 = c^2$
and:
:$\dfrac {a b} 2 = F_n \times F_{n + 1} \times F_{n + 2} \times F_{n + 3}$
That is, if the [[D... | By definition of [[Definition:Fibonacci Numbers|Fibonacci numbers]]:
:$F_n = F_{n + 2} - F_{n + 1}$
and:
:$F_{n + 3} = F_{n + 2} + F_{n + 1}$
Then $a$ can be expressed as:
{{begin-eqn}}
{{eqn | l = a
| r = \paren {F_{n + 2} - F_{n + 1} } \paren {F_{n + 2} + F_{n + 1} }
| c =
}}
{{eqn | r = {F_{n + 2} }^... | Pythagorean Triangle from Fibonacci Numbers | https://proofwiki.org/wiki/Pythagorean_Triangle_from_Fibonacci_Numbers | https://proofwiki.org/wiki/Pythagorean_Triangle_from_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Pythagorean Triangles"
] | [
"Definition:Fibonacci Number",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Term of Sequence",
"Definition:Term of Sequence",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Fibonacci Number",
"Definition:Fibonacci... | [
"Definition:Fibonacci Number",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Solutions of Pythagorean Equation",
"Honsberger's Identity",
"Definition:Area"
] |
proofwiki-12272 | If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.
Let $I$ be an ideal in $L$.
Let $F$ be a filter on $L$ such that
:$I \cap F = \O$
Then there exists a prime ideal $P$ in $L$:
$I \subseteq P$ and $P \cap F = \O$ | Define $X := \set {P \in \map {\operatorname {Ids} } L: I \subseteq P \land P \cap F = \O}$
where $\map {\operatorname {Ids} } L$ denotes set of all ideals in $L$.
By Set is Subset of Itself:
:$I \in X$
We will prove that:
:$\forall Z: Z \ne \O \land Z \subseteq X \land \paren {\forall Y_1, Y_2 \in Z: Y_1 \subseteq Y_2... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Distributive Lattice|distributive lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Let $F$ be a [[Definition:Filter|filter]] on $L$ such that
:$I \cap F = \O$
Then there exists a [[Definition:Prime Ideal (Order Theory)|prime id... | Define $X := \set {P \in \map {\operatorname {Ids} } L: I \subseteq P \land P \cap F = \O}$
where $\map {\operatorname {Ids} } L$ denotes [[Definition:Set of Sets|set]] of all [[Definition:Ideal (Order Theory)|ideals]] in $L$.
By [[Set is Subset of Itself]]:
:$I \in X$
We will prove that:
:$\forall Z: Z \ne \O \land... | If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter | https://proofwiki.org/wiki/If_Ideal_and_Filter_are_Disjoint_then_There_Exists_Prime_Ideal_Including_Ideal_and_Disjoint_from_Filter | https://proofwiki.org/wiki/If_Ideal_and_Filter_are_Disjoint_then_There_Exists_Prime_Ideal_Including_Ideal_and_Disjoint_from_Filter | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Distributive Lattice",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Prime Ideal (Order Theory)"
] | [
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Set is Subset of Itself",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Lower Section",
"Every Element is Lower implies Union is Lower",
"Definition:Lower Section",
"Definition:Non-Empty Set",
"Definition:Subset",
"Set is... |
proofwiki-12273 | No 4 Fibonacci Numbers can be in Arithmetic Sequence | Let $a, b, c, d$ be distinct Fibonacci numbers.
Then, except for the trivial case:
:$a = 0, b = 1, c = 2, d = 3$
it is not possible that $a, b, c, d$ are in arithmetic sequence. | Let:
:$a = F_i, b = F_j, c = F_k, d = F_l$
where $F_n$ denotes the $n$th Fibonacci number.
{{WLOG}}, further suppose that;
:$a < b < c < d$
or equivalently:
:$i < j < k < l$
Since $i, j, k, l$ are integers, the inequality could be written as:
:$i \le j - 1 \le k - 2 \le l - 3$
Now consider:
{{begin-eqn}}
{{eqn | l = d ... | Let $a, b, c, d$ be distinct [[Definition:Fibonacci Number|Fibonacci numbers]].
Then, except for the trivial case:
:$a = 0, b = 1, c = 2, d = 3$
it is not possible that $a, b, c, d$ are in [[Definition:Arithmetic Sequence|arithmetic sequence]]. | Let:
:$a = F_i, b = F_j, c = F_k, d = F_l$
where $F_n$ denotes the [[Definition:Fibonacci Number|$n$th Fibonacci number]].
{{WLOG}}, further suppose that;
:$a < b < c < d$
or equivalently:
:$i < j < k < l$
Since $i, j, k, l$ are [[Definition:Integer|integers]], the [[Definition:Inequality|inequality]] could be writ... | No 4 Fibonacci Numbers can be in Arithmetic Sequence | https://proofwiki.org/wiki/No_4_Fibonacci_Numbers_can_be_in_Arithmetic_Sequence | https://proofwiki.org/wiki/No_4_Fibonacci_Numbers_can_be_in_Arithmetic_Sequence | [
"Fibonacci Numbers",
"Arithmetic Sequences"
] | [
"Definition:Fibonacci Number",
"Definition:Arithmetic Sequence"
] | [
"Definition:Fibonacci Number",
"Definition:Integer",
"Definition:Inequality",
"Definition:Arithmetic Sequence",
"Definition:Fibonacci Number",
"Definition:Arithmetic Sequence",
"Definition:Fibonacci Number",
"Definition:Arithmetic Sequence",
"Definition:Fibonacci Number",
"Definition:Arithmetic Se... |
proofwiki-12274 | Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 2} \left({-1}\right)^k \dfrac 1 {F_k F_{k + 1} }
| r = \dfrac 1 {1 \times 2} - \dfrac 1 {2 \times 3} + \dfrac 1 {3 \times 5} - \dfrac 1 {5 \times 8} + \cdots
| c =
}}
{{eqn | r = \phi^{-2}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th Fibonacci nu... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }
| r = \sum_{k \mathop \ge 2} \paren {F_{k + 1} F_{k - 1} - F_k^2} \dfrac 1 {F_k F_{k + 1} }
| c = Cassini's Identity
}}
{{eqn | r = \sum_{k \mathop \ge 2} \paren {\dfrac {F_{k - 1} } {F_k} - \dfrac {F_k} {F_{k + 1} } }
... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 2} \left({-1}\right)^k \dfrac 1 {F_k F_{k + 1} }
| r = \dfrac 1 {1 \times 2} - \dfrac 1 {2 \times 3} + \dfrac 1 {3 \times 5} - \dfrac 1 {5 \times 8} + \cdots
| c =
}}
{{eqn | r = \phi^{-2}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th [[Definitio... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }
| r = \sum_{k \mathop \ge 2} \paren {F_{k + 1} F_{k - 1} - F_k^2} \dfrac 1 {F_k F_{k + 1} }
| c = [[Cassini's Identity]]
}}
{{eqn | r = \sum_{k \mathop \ge 2} \paren {\dfrac {F_{k - 1} } {F_k} - \dfrac {F_k} {F_{k + 1} ... | Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared | https://proofwiki.org/wiki/Sum_of_Alternating_Sign_Reciprocals_of_Sequence_of_Pairs_of_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean_Squared | https://proofwiki.org/wiki/Sum_of_Alternating_Sign_Reciprocals_of_Sequence_of_Pairs_of_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean_Squared | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Cassini's Identity",
"Definition:Telescoping Series",
"Ratio of Consecutive Fibonacci Numbers"
] |
proofwiki-12275 | Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }
| r = \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots
| c =
}}
{{eqn | r = \phi^{-2}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th Fibonacci number
:$\phi... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }
| r = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} } \paren {\dfrac {F_{2 k + 2} - F_{2 k} } {F_{2 k + 2} - F_{2 k} } }
| c = multiplying by $1$
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }
| r = \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots
| c =
}}
{{eqn | r = \phi^{-2}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th [[Definition:Fibonacci... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }
| r = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} } \paren {\dfrac {F_{2 k + 2} - F_{2 k} } {F_{2 k + 2} - F_{2 k} } }
| c = multiplying by $1$
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac... | Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Sequence_of_Pairs_of_Even_Index_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean_Squared | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Sequence_of_Pairs_of_Even_Index_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean_Squared | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared"
] |
proofwiki-12276 | Number of Fibonacci Numbers between n and 2n | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Then there exists either one or two Fibonacci numbers between $n$ and $2 n$ inclusive. | First existence is demonstrated.
Let $F_m \ge n$ such that $F_{m - 1} < n$.
{{begin-eqn}}
{{eqn | l = F_m
| r = F_{m - 1} + F_{m - 2}
| c = Definition of Fibonacci Numbers
}}
{{eqn | o = <
| r = 2 F_{m - 1}
| c = as $F_{m - 2} < F_{m - 1}$
}}
{{eqn | o = <
| r = 2 n
| c = as $F_{m - ... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then there exists either one or two [[Definition:Fibonacci Numbers|Fibonacci numbers]] between $n$ and $2 n$ inclusive. | First existence is demonstrated.
Let $F_m \ge n$ such that $F_{m - 1} < n$.
{{begin-eqn}}
{{eqn | l = F_m
| r = F_{m - 1} + F_{m - 2}
| c = Definition of [[Definition:Fibonacci Numbers|Fibonacci Numbers]]
}}
{{eqn | o = <
| r = 2 F_{m - 1}
| c = as $F_{m - 2} < F_{m - 1}$
}}
{{eqn | o = <
... | Number of Fibonacci Numbers between n and 2n | https://proofwiki.org/wiki/Number_of_Fibonacci_Numbers_between_n_and_2n | https://proofwiki.org/wiki/Number_of_Fibonacci_Numbers_between_n_and_2n | [
"Fibonacci Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Fibonacci Number"
] | [
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:By Hypothesis",
"Proof by Contradiction",
"Definition:Fibonacci Number",
"Definition:Fibonacci Numbe... |
proofwiki-12277 | Square Root is of Exponential Order Epsilon | The positive square root function:
:$t \mapsto \sqrt t$
is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | {{begin-eqn}}
{{eqn | l = \sqrt t
| o = <
| r = K e^{a t}
| c = an Ansatz
}}
{{eqn | ll = \impliedby
| l = t
| o = <
| r = \paren {K e^{a t} }^2
| c = Square Root is Strictly Increasing
}}
{{eqn | r = K^2 e^{2 a t}
| c = Exponential of Product
}}
{{eqn | r = K' e^{a' t}
... | The [[Definition:Positive Square Root|positive square root function]]:
:$t \mapsto \sqrt t$
is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | {{begin-eqn}}
{{eqn | l = \sqrt t
| o = <
| r = K e^{a t}
| c = an [[Definition:Ansatz|Ansatz]]
}}
{{eqn | ll = \impliedby
| l = t
| o = <
| r = \paren {K e^{a t} }^2
| c = [[Square Root is Strictly Increasing]]
}}
{{eqn | r = K^2 e^{2 a t}
| c = [[Exponential of Product]... | Square Root is of Exponential Order Epsilon | https://proofwiki.org/wiki/Square_Root_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Square_Root_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Square Root/Positive",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Ansatz",
"Square Root is Strictly Increasing",
"Exponential of Product",
"Identity is of Exponential Order Epsilon",
"Category:Exponential Order"
] |
proofwiki-12278 | Number of Fibonacci Numbers with Same Number of Decimal Digits | Let $n$ be an integer such that $n > 1$.
When expressed in decimal notation, there are either $4$ or $5$ Fibonacci numbers with $n$ digits. | Let $F_m$ be an $n$-digit Fibonacci number.
Then $F_m \ge 10^{n - 1} \ge 10$.
We have:
{{begin-eqn}}
{{eqn | l = F_m
| r = F_{m - 1} + F_{m - 2}
| c = {{Defof|Fibonacci Numbers}}
}}
{{eqn | o = \le
| r = 2 F_{m - 1}
| c = as $F_{m - 2} \le F_{m - 1}$
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l ... | Let $n$ be an [[Definition:Integer|integer]] such that $n > 1$.
When expressed in [[Definition:Decimal Notation|decimal notation]], there are either $4$ or $5$ [[Definition:Fibonacci Number|Fibonacci numbers]] with $n$ [[Definition:Digit|digits]]. | Let $F_m$ be an $n$-[[Definition:Digit|digit]] [[Definition:Fibonacci Number|Fibonacci number]].
Then $F_m \ge 10^{n - 1} \ge 10$.
We have:
{{begin-eqn}}
{{eqn | l = F_m
| r = F_{m - 1} + F_{m - 2}
| c = {{Defof|Fibonacci Numbers}}
}}
{{eqn | o = \le
| r = 2 F_{m - 1}
| c = as $F_{m - 2} \le ... | Number of Fibonacci Numbers with Same Number of Decimal Digits | https://proofwiki.org/wiki/Number_of_Fibonacci_Numbers_with_Same_Number_of_Decimal_Digits | https://proofwiki.org/wiki/Number_of_Fibonacci_Numbers_with_Same_Number_of_Decimal_Digits | [
"Fibonacci Numbers"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Fibonacci Number",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Fibonacci Number",
"Honsberger's Identity",
"Definition:Digit",
"Definition:Fibonacci Number",
"Definition:Digit",
"Definition:Fibonacci Number",
"Definition:Digit",
"Honsberger's Identity",
"Definition:Fibonacci Number",
"Definition:Digit",
"Definition:Fibonacc... |
proofwiki-12279 | Prime Number divides Infinite Number of Fibonacci Numbers | Let $p$ be a prime number.
Then there exist an infinite number of Fibonacci numbers which are divisible by $p$. | From Prime Number divides Fibonacci Number, either $F_{p - 1}$ or $F_{p + 1}$ is divisible by $p$.
Thus:
{{begin-eqn}}
{{eqn | l = p
| o = \divides
| r = F_{p \pm 1}
| c = Prime Number divides Fibonacci Number
}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_{p \pm 1}
| o = \divides
| r ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then there exist an [[Definition:Infinite Set|infinite number]] of [[Definition:Fibonacci Number|Fibonacci numbers]] which are [[Definition:Divisor of Integer|divisible]] by $p$. | From [[Prime Number divides Fibonacci Number]], either $F_{p - 1}$ or $F_{p + 1}$ is [[Definition:Divisor of Integer|divisible]] by $p$.
Thus:
{{begin-eqn}}
{{eqn | l = p
| o = \divides
| r = F_{p \pm 1}
| c = [[Prime Number divides Fibonacci Number]]
}}
{{eqn | q = \forall n \in \Z_{>0}
| l = ... | Prime Number divides Infinite Number of Fibonacci Numbers | https://proofwiki.org/wiki/Prime_Number_divides_Infinite_Number_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Prime_Number_divides_Infinite_Number_of_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Divisors"
] | [
"Definition:Prime Number",
"Definition:Infinite Set",
"Definition:Fibonacci Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Prime Number divides Fibonacci Number",
"Definition:Divisor (Algebra)/Integer",
"Prime Number divides Fibonacci Number",
"Divisibility of Fibonacci Number"
] |
proofwiki-12280 | Prime Number divides Fibonacci Number | For $n \in \Z$, let $F_n$ denote the $n$th Fibonacci number.
Let $p$ be a prime number.
Then:
:$p \equiv \pm 1 \pmod 5 \implies p \divides F_{p - 1}$
:$p \equiv \pm 2 \pmod 5 \implies p \divides F_{p + 1}$
where $\divides$ denotes divisibility.
Thus in all cases, except where $p = 5$ itself:
:$p \divides F_{p \pm 1}$ | It is worth noting the one case where $p = 5$:
:$5 \divides F_5 = 5$
{{ProofWanted|Googling around suggests there is a proof based on the Law of Quadratic Reciprocity but I have not laid hands on it yet.}} | For $n \in \Z$, let $F_n$ denote the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$p \equiv \pm 1 \pmod 5 \implies p \divides F_{p - 1}$
:$p \equiv \pm 2 \pmod 5 \implies p \divides F_{p + 1}$
where $\divides$ denotes [[Definition:Divisor of In... | It is worth noting the one case where $p = 5$:
:$5 \divides F_5 = 5$
{{ProofWanted|Googling around suggests there is a proof based on the [[Law of Quadratic Reciprocity]] but I have not laid hands on it yet.}} | Prime Number divides Fibonacci Number | https://proofwiki.org/wiki/Prime_Number_divides_Fibonacci_Number | https://proofwiki.org/wiki/Prime_Number_divides_Fibonacci_Number | [
"Fibonacci Numbers",
"Divisors"
] | [
"Definition:Fibonacci Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Law of Quadratic Reciprocity"
] |
proofwiki-12281 | Existence of Fibonacci Number Divisible by Number | Let $m \in \Z_{\ne 0}$ be an integer.
Then in the first $m^2$ Fibonacci numbers there exists at least one Fibonacci number which is divisible by $m$. | Consider pairs of Fibonacci numbers $\tuple {F_i, F_{i + 1}}$ modulo $m$.
There are $m^2$ possible pairs of remainders.
Thus by Pigeonhole Principle, among the $\paren {m^2 + 1}$ pairs of $\tuple {F_i, F_{i + 1}}$ with $0 \le i \le m^2$, at least two of them are identical modulo $m$.
That is, there exists $0 \le i < j ... | Let $m \in \Z_{\ne 0}$ be an [[Definition:Integer|integer]].
Then in the first $m^2$ [[Definition:Fibonacci Numbers|Fibonacci numbers]] there exists at least one [[Definition:Fibonacci Numbers|Fibonacci number]] which is [[Definition:Divisor of Integer|divisible]] by $m$. | Consider pairs of [[Definition:Fibonacci Numbers|Fibonacci numbers]] $\tuple {F_i, F_{i + 1}}$ [[Definition:Modulo Arithmetic|modulo]] $m$.
There are $m^2$ possible pairs of remainders.
Thus by [[Pigeonhole Principle]], among the $\paren {m^2 + 1}$ pairs of $\tuple {F_i, F_{i + 1}}$ with $0 \le i \le m^2$, at least t... | Existence of Fibonacci Number Divisible by Number | https://proofwiki.org/wiki/Existence_of_Fibonacci_Number_Divisible_by_Number | https://proofwiki.org/wiki/Existence_of_Fibonacci_Number_Divisible_by_Number | [
"Fibonacci Numbers",
"Divisors"
] | [
"Definition:Integer",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Fibonacci Number",
"Definition:Modulo Arithmetic",
"Dirichlet's Box Principle/Corollary",
"Definition:Modulo Arithmetic",
"Principle of Mathematical Induction",
"Definition:Integer"
] |
proofwiki-12282 | Fibonacci Prime has Prime Index except for 3 | Let $F_n$ denote the $n$th Fibonacci number.
Let $F_n$ be a prime number.
Then, apart from $F_4 = 3$, $n$ is a prime number. | Let $F_n$ be a prime number.
{{AimForCont}} $n$ is a composite number greater than $4$.
Then $n = r s$ for some $1 < r, s < n$.
Except for the case where $n = 4$, at least one of $r$ and $s$ is greater than $2$.
From Divisibility of Fibonacci Number:
:$F_r \divides F_n$
and:
:$F_s \divides F_n$
where $\divides$ denotes... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Let $F_n$ be a [[Definition:Prime Number|prime number]].
Then, apart from $F_4 = 3$, $n$ is a [[Definition:Prime Number|prime number]]. | Let $F_n$ be a [[Definition:Prime Number|prime number]].
{{AimForCont}} $n$ is a [[Definition:Composite Number|composite number]] greater than $4$.
Then $n = r s$ for some $1 < r, s < n$.
Except for the case where $n = 4$, at least one of $r$ and $s$ is greater than $2$.
From [[Divisibility of Fibonacci Number]]:
:... | Fibonacci Prime has Prime Index except for 3 | https://proofwiki.org/wiki/Fibonacci_Prime_has_Prime_Index_except_for_3 | https://proofwiki.org/wiki/Fibonacci_Prime_has_Prime_Index_except_for_3 | [
"Fibonacci Numbers",
"Prime Numbers",
"Fibonacci Primes"
] | [
"Definition:Fibonacci Number",
"Definition:Prime Number",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Composite Number",
"Divisibility of Fibonacci Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Composite Number",
"Definition:Proper Divisor/Integer",
"Definition:Prime Number",
"Proof by Contradiction",
"Definition:Composite Number",
"Definition:... |
proofwiki-12283 | Fibonacci Number with Prime Index is not necessarily Prime | Let $p \in \Z_{>0}$ be a prime number.
Let $F_p$ be the $p$th Fibonacci number.
Then $F_p$ is not itself necessarily prime. | Proof by Counterexample:
:$F_{19} = 4181 = 37 \times 113$
{{qed}} | Let $p \in \Z_{>0}$ be a [[Definition:Prime Number|prime number]].
Let $F_p$ be the $p$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then $F_p$ is not itself necessarily [[Definition:Prime Number|prime]]. | [[Proof by Counterexample]]:
:$F_{19} = 4181 = 37 \times 113$
{{qed}} | Fibonacci Number with Prime Index is not necessarily Prime | https://proofwiki.org/wiki/Fibonacci_Number_with_Prime_Index_is_not_necessarily_Prime | https://proofwiki.org/wiki/Fibonacci_Number_with_Prime_Index_is_not_necessarily_Prime | [
"Fibonacci Numbers",
"Prime Numbers"
] | [
"Definition:Prime Number",
"Definition:Fibonacci Number",
"Definition:Prime Number"
] | [
"Proof by Counterexample"
] |
proofwiki-12284 | Complex Riemann Integral is Contour Integral | Let $f: \R \to \C$ be a complex Riemann integrable function over some closed real interval $\closedint a b$.
Then:
:$\ds \int_a^b \map f t \rd t = \int_\CC \map f t \rd t$
where:
:the integral on the {{LHS}} is a complex Riemann integral
:the integral on the {{RHS}} is a contour integral
:$\CC$ is a straight line segme... | {{begin-eqn}}
{{eqn | l = \int_a^b \map f t \rd t
| r = \int_a^b \map f {\map \theta t} \map {\theta'} t \rd t
| c = Complex Integration by Substitution: $\map \theta t = t$, $\map {\theta'} t = 1$
}}
{{eqn | r = \int_\CC \map f t \rd t
| c = {{Defof|Complex Contour Integral}}
}}
{{end-eqn}}
{{qed}}
C... | Let $f: \R \to \C$ be a [[Definition:Complex Riemann Integrable Function|complex Riemann integrable function]] over some [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Then:
:$\ds \int_a^b \map f t \rd t = \int_\CC \map f t \rd t$
where:
:the integral on the {{LHS}} is a [[Definition:Comp... | {{begin-eqn}}
{{eqn | l = \int_a^b \map f t \rd t
| r = \int_a^b \map f {\map \theta t} \map {\theta'} t \rd t
| c = [[Complex Integration by Substitution]]: $\map \theta t = t$, $\map {\theta'} t = 1$
}}
{{eqn | r = \int_\CC \map f t \rd t
| c = {{Defof|Complex Contour Integral}}
}}
{{end-eqn}}
{{qed... | Complex Riemann Integral is Contour Integral | https://proofwiki.org/wiki/Complex_Riemann_Integral_is_Contour_Integral | https://proofwiki.org/wiki/Complex_Riemann_Integral_is_Contour_Integral | [
"Integral Calculus",
"Complex Contour Integrals"
] | [
"Definition:Integrable Function/Complex",
"Definition:Real Interval/Closed",
"Definition:Integrable Function/Complex",
"Definition:Contour Integral/Complex",
"Definition:Line/Straight Line Segment",
"Definition:Complex Number/Complex Plane/Real Axis"
] | [
"Complex Integration by Substitution",
"Category:Integral Calculus",
"Category:Complex Contour Integrals"
] |
proofwiki-12285 | Fibonacci Number is not Product of Two Smaller Fibonacci Numbers | Let $m, n \in \Z$ be integers.
Suppose $\size m, \size n \ge 3$.
Let $F_m$ and $F_n$ be the $m$th and $n$th Fibonacci numbers.
Then $F_m \times F_n$ is not a Fibonacci number. | From Honsberger's Identity:
:$F_n = F_{k - 1} F_{n - k + 2} + F_{k - 2} F_{n - k + 1}$
for $2 \le k \le n$.
{{AimForCont}} $F_n = F_m F_k$ for some $m, k \ge 3$.
Then:
{{begin-eqn}}
{{eqn | l = F_m
| r = \dfrac {F_n} {F_k}
| c =
}}
{{eqn | r = \dfrac {F_{k - 1} F_{n - k + 2} + F_{k - 2} F_{n - k + 1} } {F_... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Suppose $\size m, \size n \ge 3$.
Let $F_m$ and $F_n$ be the $m$th and $n$th [[Definition:Fibonacci Number|Fibonacci numbers]].
Then $F_m \times F_n$ is not a [[Definition:Fibonacci Number|Fibonacci number]]. | From [[Honsberger's Identity]]:
:$F_n = F_{k - 1} F_{n - k + 2} + F_{k - 2} F_{n - k + 1}$
for $2 \le k \le n$.
{{AimForCont}} $F_n = F_m F_k$ for some $m, k \ge 3$.
Then:
{{begin-eqn}}
{{eqn | l = F_m
| r = \dfrac {F_n} {F_k}
| c =
}}
{{eqn | r = \dfrac {F_{k - 1} F_{n - k + 2} + F_{k - 2} F_{n - k + ... | Fibonacci Number is not Product of Two Smaller Fibonacci Numbers | https://proofwiki.org/wiki/Fibonacci_Number_is_not_Product_of_Two_Smaller_Fibonacci_Numbers | https://proofwiki.org/wiki/Fibonacci_Number_is_not_Product_of_Two_Smaller_Fibonacci_Numbers | [
"Fibonacci Numbers"
] | [
"Definition:Integer",
"Definition:Fibonacci Number",
"Definition:Fibonacci Number"
] | [
"Honsberger's Identity",
"Definition:Weighted Mean",
"Definition:Fibonacci Number",
"Proof by Contradiction"
] |
proofwiki-12286 | Transformation of P-Norm | Let $p, q \ge 1$ be real numbers.
Let ${\ell^p}_\R$ denote the $p$-sequence space on $\R$.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
Let $\mathbf x = \sequence {x_n} \in {\ell^{p q} }_\R$.
Suppose further that $\mathbf x^p = \sequence { {x_n}^p} \in {\ell^q}_\R$.
Then:
:$\norm {\mathbf x^p}_q = \par... | {{begin-eqn}}
{{eqn | l = \norm {\mathbf x^p}_q
| r = \paren {\sum_{n \mathop = 0}^\infty \size { {x_n}^p}^q}^{1 / q}
| c = {{Defof|P-Norm|$p$-Norm}}
}}
{{eqn | r = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^{p q} }^{1 / q}
| c = Power of Power
}}
{{eqn | r = \paren {\paren {\sum_{n \mathop = 0}^... | Let $p, q \ge 1$ be [[Definition:Real Number|real numbers]].
Let ${\ell^p}_\R$ denote the [[Definition:Real P-Sequence Space|$p$-sequence space on $\R$]].
Let $\norm {\mathbf x}_p$ denote the [[Definition:Real P-Norm|$p$-norm]] of $\mathbf x$.
Let $\mathbf x = \sequence {x_n} \in {\ell^{p q} }_\R$.
Suppose further... | {{begin-eqn}}
{{eqn | l = \norm {\mathbf x^p}_q
| r = \paren {\sum_{n \mathop = 0}^\infty \size { {x_n}^p}^q}^{1 / q}
| c = {{Defof|P-Norm|$p$-Norm}}
}}
{{eqn | r = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^{p q} }^{1 / q}
| c = [[Power of Power]]
}}
{{eqn | r = \paren {\paren {\sum_{n \mathop =... | Transformation of P-Norm | https://proofwiki.org/wiki/Transformation_of_P-Norm | https://proofwiki.org/wiki/Transformation_of_P-Norm | [
"Functional Analysis",
"Norm Theory",
"P-Sequence Metrics",
"P-Norms"
] | [
"Definition:Real Number",
"Definition:P-Sequence Space/Real",
"Definition:P-Norm/Real"
] | [
"Exponent Combination Laws/Power of Power",
"Exponent Combination Laws/Power of Power",
"Category:Functional Analysis",
"Category:Norm Theory",
"Category:P-Sequence Metrics",
"Category:P-Norms"
] |
proofwiki-12287 | Set is Subset of Finite Suprema Set | Let $\struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then $X \subseteq \map {\mathrm {finsups} } X$
where $\map {\mathrm {finsups} } X$ denotes finite suprema set of $X$. | Let $x \in X$.
By Supremum of Singleton:
:$\set x$ admits a supremum and $\sup \set x = x$
By definitions of subset and singleton:
:$\set x \subseteq X$
By Singleton is Finite:
:$\set x$ is a finite set.
Thus by definition of finite suprema set:
:$x \in \map {\mathrm {finsups} } X$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then $X \subseteq \map {\mathrm {finsups} } X$
where $\map {\mathrm {finsups} } X$ denotes [[Definition:Finite Suprema Set|finite suprema set]] of $X$. | Let $x \in X$.
By [[Supremum of Singleton]]:
:$\set x$ admits a [[Definition:Supremum of Set|supremum]] and $\sup \set x = x$
By definitions of [[Definition:Subset|subset]] and [[Definition:Singleton|singleton]]:
:$\set x \subseteq X$
By [[Singleton is Finite]]:
:$\set x$ is a [[Definition:Finite Set|finite set]].
... | Set is Subset of Finite Suprema Set | https://proofwiki.org/wiki/Set_is_Subset_of_Finite_Suprema_Set | https://proofwiki.org/wiki/Set_is_Subset_of_Finite_Suprema_Set | [
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Finite Suprema Set"
] | [
"Supremum of Singleton",
"Definition:Supremum of Set",
"Definition:Subset",
"Definition:Singleton",
"Singleton is Finite",
"Definition:Finite Set",
"Definition:Finite Suprema Set"
] |
proofwiki-12288 | Lower Closure of Subset is Subset of Lower Closure | Let $\struct {S, \preceq}$ be an ordered set.
Let $X, Y$ be subsets of $S$.
Then
:$X \subseteq Y \implies X^\preceq \subseteq Y^\preceq$
where $X^\preceq$ is the lower closure of $X$. | Let $X \subseteq Y$.
Let $x \in X^\preceq$.
By definition of lower closure of subset:
:$\exists y \in X: x \preceq y$
By definition of subset:
:$y \in Y$
Thus by definition of lower closure of subset:
:$x \in Y^\preceq$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X, Y$ be [[Definition:Subset|subsets]] of $S$.
Then
:$X \subseteq Y \implies X^\preceq \subseteq Y^\preceq$
where $X^\preceq$ is the [[Definition:Lower Closure of Subset|lower closure]] of $X$. | Let $X \subseteq Y$.
Let $x \in X^\preceq$.
By definition of [[Definition:Lower Closure of Subset|lower closure of subset]]:
:$\exists y \in X: x \preceq y$
By definition of [[Definition:Subset|subset]]:
:$y \in Y$
Thus by definition of [[Definition:Lower Closure of Subset|lower closure of subset]]:
:$x \in Y^\p... | Lower Closure of Subset is Subset of Lower Closure | https://proofwiki.org/wiki/Lower_Closure_of_Subset_is_Subset_of_Lower_Closure | https://proofwiki.org/wiki/Lower_Closure_of_Subset_is_Subset_of_Lower_Closure | [
"Order Theory",
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Lower Closure/Set"
] | [
"Definition:Lower Closure/Set",
"Definition:Subset",
"Definition:Lower Closure/Set"
] |
proofwiki-12289 | Finite Suprema Set and Lower Closure is Smallest Ideal | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $X$ be a subset of $S$.
Then $X \subseteq \map {\operatorname {finsups} } X^\preceq$ and
:for every ideal $I$ in $L$: $X \subseteq I \implies \map {\operatorname {finsups} } X^\preceq \subseteq I$
where
:$\map {\operatorname {finsups} } X$ denotes the fini... | By Set is Subset of Finite Suprema Set:
:$X \subseteq \map {\operatorname {finsups} } X$
By Lower Closure of Subset is Subset of Lower Closure:
:$X^\preceq \subseteq \map {\operatorname {finsups} } X^\preceq$
By Set is Subset of Lower Closure:
:$X \subseteq X^\preceq$
Thus by Subset Relation is Transitive:
:$X \subsete... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then $X \subseteq \map {\operatorname {finsups} } X^\preceq$ and
:for every [[Definition:Ideal (Order Theory)|ideal]] $I$ in $L$: $X \subseteq I \implies \map {\operatorname {... | By [[Set is Subset of Finite Suprema Set]]:
:$X \subseteq \map {\operatorname {finsups} } X$
By [[Lower Closure of Subset is Subset of Lower Closure]]:
:$X^\preceq \subseteq \map {\operatorname {finsups} } X^\preceq$
By [[Set is Subset of Lower Closure]]:
:$X \subseteq X^\preceq$
Thus by [[Subset Relation is Transit... | Finite Suprema Set and Lower Closure is Smallest Ideal | https://proofwiki.org/wiki/Finite_Suprema_Set_and_Lower_Closure_is_Smallest_Ideal | https://proofwiki.org/wiki/Finite_Suprema_Set_and_Lower_Closure_is_Smallest_Ideal | [
"Order Theory",
"Suprema"
] | [
"Definition:Join Semilattice",
"Definition:Subset",
"Definition:Ideal (Order Theory)",
"Definition:Finite Suprema Set",
"Definition:Lower Closure/Set"
] | [
"Set is Subset of Finite Suprema Set",
"Lower Closure of Subset is Subset of Lower Closure",
"Set is Subset of Lower Closure",
"Subset Relation is Transitive",
"Definition:Ideal (Order Theory)",
"Definition:Lower Closure/Set",
"Definition:Finite Suprema Set",
"Definition:Supremum of Set",
"Definitio... |
proofwiki-12290 | Set is Subset of Lower Closure | Let $\struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then $X \subseteq X^\preceq$
where $X^\preceq$ denotes the lower closure of $X$. | Let $x \in X$.
By definition of reflexivity:
:$x \preceq x$
Thus by definition of lower closure:
:$x \in X^\preceq$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then $X \subseteq X^\preceq$
where $X^\preceq$ denotes the [[Definition:Lower Closure of Subset|lower closure]] of $X$. | Let $x \in X$.
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \preceq x$
Thus by definition of [[Definition:Lower Closure of Subset|lower closure]]:
:$x \in X^\preceq$
{{qed}} | Set is Subset of Lower Closure | https://proofwiki.org/wiki/Set_is_Subset_of_Lower_Closure | https://proofwiki.org/wiki/Set_is_Subset_of_Lower_Closure | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Lower Closure/Set"
] | [
"Definition:Reflexivity",
"Definition:Lower Closure/Set"
] |
proofwiki-12291 | Maximum Volume of Unit Radius Sphere in Fractional Dimensions | The maximum volume of a unit sphere in $x$-dimensional Euclidean space for real $x$ occurs when $x$ is given as:
:$x = 5 \cdotp 25694 \, 64048 \, 60 \ldots$
{{OEIS|A074455}}
The corresponding volume at that dimension is given by:
:$V = 5 \cdotp 27776 \, 80211 \, 13400 \, 997 \ldots$
{{OEIS|A074454}} | {{ProofWanted|Requires significant work to even define the concepts.}} | The [[Definition:Maximum Value|maximum]] [[Definition:Volume|volume]] of a [[Definition:Unit Sphere (Topology)|unit sphere]] in [[Definition:Dimension (Geometry)|$x$-dimensional]] [[Definition:Euclidean Space|Euclidean space]] for [[Definition:Real Number|real]] $x$ occurs when $x$ is given as:
:$x = 5 \cdotp 25694 \, ... | {{ProofWanted|Requires significant work to even define the concepts.}} | Maximum Volume of Unit Radius Sphere in Fractional Dimensions | https://proofwiki.org/wiki/Maximum_Volume_of_Unit_Radius_Sphere_in_Fractional_Dimensions | https://proofwiki.org/wiki/Maximum_Volume_of_Unit_Radius_Sphere_in_Fractional_Dimensions | [
"Geometry"
] | [
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Volume",
"Definition:Unit Sphere/Topology",
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space",
"Definition:Real Number",
"Definition:Volume",
"Definition:Dimension (Geometry)"
] | [] |
proofwiki-12292 | Even Perfect Number is Sum of Successive Odd Cubes except 6 | Let $n$ be an even perfect number such that $n \ne 6$.
Then:
:$\ds n = \sum_{k \mathop = 1}^m \paren {2 k - 1}^3 = 1^3 + 3^3 + \cdots + \paren {2 m - 1}^3$
for some $m \in \Z_{>0}$.
That is, every even perfect number apart from $6$ is the sum of the sequence of the first $m$ odd cubes, for some $m$. | From Sum of Sequence of Odd Cubes:
:$1^3 + 3^3 + 5^3 + \cdots + \paren {2 m − 1}^3 = m^2 \paren {2 m^2 − 1}$
By the Theorem of Even Perfect Numbers:
:$n = 2^{p - 1} \paren {2^p - 1}$
for some $p$, where $2^p - 1$ is prime.
From Primes of form Power Less One, it follows that $p$ is itself prime.
Let $p$ be an odd prime.... | Let $n$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]] such that $n \ne 6$.
Then:
:$\ds n = \sum_{k \mathop = 1}^m \paren {2 k - 1}^3 = 1^3 + 3^3 + \cdots + \paren {2 m - 1}^3$
for some $m \in \Z_{>0}$.
That is, every [[Definition:Even Integer|even]] [[Definition:Perfect Number|... | From [[Sum of Sequence of Odd Cubes]]:
:$1^3 + 3^3 + 5^3 + \cdots + \paren {2 m − 1}^3 = m^2 \paren {2 m^2 − 1}$
By the [[Theorem of Even Perfect Numbers]]:
:$n = 2^{p - 1} \paren {2^p - 1}$
for some $p$, where $2^p - 1$ is [[Definition:Prime Number|prime]].
From [[Primes of form Power Less One]], it follows that $p$... | Even Perfect Number is Sum of Successive Odd Cubes except 6 | https://proofwiki.org/wiki/Even_Perfect_Number_is_Sum_of_Successive_Odd_Cubes_except_6 | https://proofwiki.org/wiki/Even_Perfect_Number_is_Sum_of_Successive_Odd_Cubes_except_6 | [
"Euclidean Numbers",
"Perfect Numbers",
"Sums of Sequences",
"Cube Numbers"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Sequence",
"Definition:Odd Integer",
"Definition:Cube Number"
] | [
"Sum of Sequence of Odd Cubes",
"Theorem of Even Perfect Numbers",
"Definition:Prime Number",
"Primes of form Power Less One",
"Definition:Prime Number",
"Definition:Odd Prime",
"Definition:Even Integer",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Prime Number",
"Definition:Per... |
proofwiki-12293 | Rational Power is of Exponential Order Epsilon | Let $r = \dfrac p q$ be a rational number, with $p, q \in \Z: q \ne 0, r > 0$.
Then:
:$t \mapsto t^r$
is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | Write $t^r = t^{p/q}$, and set $t > 1$.
{{begin-eqn}}
{{eqn | l = t^{p/q}
| o = <
| r = K e^{a t}
| c = an Ansatz
}}
{{eqn | ll = \impliedby
| l = t^p
| o = <
| r = \paren {K e^{a t} }^q
| c = Rational Power is Strictly Increasing
}}
{{eqn | r = K^q e^{q a t}
| c = Expon... | Let $r = \dfrac p q$ be a [[Definition:Rational Number|rational number]], with $p, q \in \Z: q \ne 0, r > 0$.
Then:
:$t \mapsto t^r$
is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | Write $t^r = t^{p/q}$, and set $t > 1$.
{{begin-eqn}}
{{eqn | l = t^{p/q}
| o = <
| r = K e^{a t}
| c = an [[Definition:Ansatz|Ansatz]]
}}
{{eqn | ll = \impliedby
| l = t^p
| o = <
| r = \paren {K e^{a t} }^q
| c = [[Power Function on Base Greater than One is Strictly Increas... | Rational Power is of Exponential Order Epsilon | https://proofwiki.org/wiki/Rational_Power_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Rational_Power_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Rational Number",
"Definition:Exponential Order/Real Index"
] | [
"Definition:Ansatz",
"Power Function on Base Greater than One is Strictly Increasing/Rational Number",
"Exponential of Product",
"Polynomial is of Exponential Order Epsilon",
"Category:Exponential Order"
] |
proofwiki-12294 | Sum of Sequence of Odd Cubes | :$\ds \sum_{r \mathop = 1}^n \paren {2 r - 1}^3 = 1^3 + 3^3 + 5^3 + \dotsb + \paren {2 n − 1}^3 = n^2 \paren {2 n^2 − 1}$ | {{begin-eqn}}
{{eqn | n = d
| l = S_n
| o = :=
| r = \sum_{r \mathop = 1}^n r^3
| c =
}}
{{eqn | r = 1^3 + 2^3 + 3^3 + \cdots + \paren {n - 1}^3 + n^3
| c =
}}
{{eqn | n = d
| l = O_n
| o = :=
| r = \sum_{r \mathop = 1}^n \paren {2 r - 1}^3
| c =
}}
{{eqn | r = 1... | :$\ds \sum_{r \mathop = 1}^n \paren {2 r - 1}^3 = 1^3 + 3^3 + 5^3 + \dotsb + \paren {2 n − 1}^3 = n^2 \paren {2 n^2 − 1}$ | {{begin-eqn}}
{{eqn | n = d
| l = S_n
| o = :=
| r = \sum_{r \mathop = 1}^n r^3
| c =
}}
{{eqn | r = 1^3 + 2^3 + 3^3 + \cdots + \paren {n - 1}^3 + n^3
| c =
}}
{{eqn | n = d
| l = O_n
| o = :=
| r = \sum_{r \mathop = 1}^n \paren {2 r - 1}^3
| c =
}}
{{eqn | r = 1... | Sum of Sequence of Odd Cubes/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Cubes | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Cubes/Proof_1 | [
"Sum of Sequence of Odd Cubes",
"Sums of Sequences",
"Cube Numbers"
] | [] | [
"Sum of Sequence of Cubes"
] |
proofwiki-12295 | Bottom in Ideal | Let $\struct {S, \preceq}$ be a bounded below ordered set.
Let $I$ be a ideal in $S$.
Then $\bot \in I$
where $\bot$ denotes the smallest element of $S$. | By definition of ideal in ordered set:
:$I$ is non-empty and lower.
By definition of non-empty set:
:$\exists x: x \in I$
By definition of smallest element:
:$\bot \preceq x$
Thus by definition of lower section:
:$\bot \in I$
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Ordered Set|ordered set]].
Let $I$ be a [[Definition:Ideal (Order Theory)|ideal]] in $S$.
Then $\bot \in I$
where $\bot$ denotes the [[Definition:Smallest Element|smallest element]] of $S$. | By definition of [[Definition:Ideal in Ordered Set|ideal in ordered set]]:
:$I$ is [[Definition:Non-Empty Set|non-empty]] and [[Definition:Lower Section|lower]].
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists x: x \in I$
By definition of [[Definition:Smallest Element|smallest element]]:
:$\bot... | Bottom in Ideal | https://proofwiki.org/wiki/Bottom_in_Ideal | https://proofwiki.org/wiki/Bottom_in_Ideal | [
"Order Theory"
] | [
"Definition:Bounded Below Set",
"Definition:Ordered Set",
"Definition:Ideal (Order Theory)",
"Definition:Smallest Element"
] | [
"Definition:Ideal in Ordered Set",
"Definition:Non-Empty Set",
"Definition:Lower Section",
"Definition:Non-Empty Set",
"Definition:Smallest Element",
"Definition:Lower Section"
] |
proofwiki-12296 | Real Power is of Exponential Order Epsilon | Let:
:$f: \hointr 0 \to \to \R: t \mapsto t^r$
be $t$ to the power of $r$, for $r \in \R, r > -1$.
Then $f$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude. | For $t > 0$, $t^r$ is continuous.
At $t = 0$, defining $0^r = 0$, the function is continuous from the right:
{{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to 0^+} t^r
| r = \lim_{t \mathop \to 0^+} \map \exp {r \ln t}
}}
{{eqn | r = \map \exp {\lim_{t \mathop \to 0^+} r \ln t}
| c = Exponential Function is Con... | Let:
:$f: \hointr 0 \to \to \R: t \mapsto t^r$
be [[Definition:Power (Algebra)/Real Number/Definition 1|$t$ to the power of $r$]], for $r \in \R, r > -1$.
Then $f$ is of [[Definition:Exponential Order to Real Index|exponential order $\epsilon$]] for any $\epsilon > 0$ arbitrarily small in magnitude. | For $t > 0$, [[Power Function on Strictly Positive Base is Continuous/Rational Power|$t^r$ is continuous]].
At $t = 0$, [[Definition:Power of Zero|defining $0^r = 0$]], the function is [[Definition:Right-Continuous at Point|continuous from the right]]:
{{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to 0^+} t^r
| r ... | Real Power is of Exponential Order Epsilon | https://proofwiki.org/wiki/Real_Power_is_of_Exponential_Order_Epsilon | https://proofwiki.org/wiki/Real_Power_is_of_Exponential_Order_Epsilon | [
"Exponential Order"
] | [
"Definition:Power (Algebra)/Real Number/Definition 1",
"Definition:Exponential Order/Real Index"
] | [
"Power Function on Strictly Positive Base is Continuous/Rational Power",
"Definition:Power (Algebra)/Power of Zero",
"Definition:Continuous Real Function/Right-Continuous",
"Exponential Function is Continuous/Real Numbers",
"Logarithm Tends to Negative Infinity",
"Exponential Tends to Zero and Infinity",
... |
proofwiki-12297 | Finite Suprema Set and Lower Closure is Ideal | Let $P = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $X$ be a non-empty subset of $S$.
Then
:$\map {\operatorname{finsups} } X^\preceq$ is ideal in $P$.
where
:$\map {\operatorname{finsups} } X$ denotes the finite suprema set of $X$,
:$X^\preceq$ denotes the lower closure of $X$. | By Finite Suprema Set and Lower Closure is Smallest Ideal:
:$X \subseteq \map {\operatorname{finsups} } X^\preceq$
By definition of non-empty set:
:$\map {\operatorname{finsups} } X^\preceq$ is a non-empty set.
We will prove that
:$\map {\operatorname{finsups} } X$ is directed.
Let $x, y \in \map {\operatorname{fininfs... | Let $P = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then
:$\map {\operatorname{finsups} } X^\preceq$ is [[Definition:Ideal (Order Theory)|ideal]] in $P$.
where
:$\map {\operatorname{finsup... | By [[Finite Suprema Set and Lower Closure is Smallest Ideal]]:
:$X \subseteq \map {\operatorname{finsups} } X^\preceq$
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\map {\operatorname{finsups} } X^\preceq$ is a [[Definition:Non-Empty Set|non-empty set]].
We will prove that
:$\map {\operatorname{fins... | Finite Suprema Set and Lower Closure is Ideal | https://proofwiki.org/wiki/Finite_Suprema_Set_and_Lower_Closure_is_Ideal | https://proofwiki.org/wiki/Finite_Suprema_Set_and_Lower_Closure_is_Ideal | [
"Order Theory",
"Suprema"
] | [
"Definition:Join Semilattice",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Ideal (Order Theory)",
"Definition:Finite Suprema Set",
"Definition:Lower Closure/Set"
] | [
"Finite Suprema Set and Lower Closure is Smallest Ideal",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Directed Subset",
"Definition:Finite Suprema Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Set of Sets",
"Definition:Finite Set",
"Definitio... |
proofwiki-12298 | Set of 3 Integers each Divisor of Sum of Other Two | There exists exactly one set of distinct coprime positive integers such that each is a divisor of the sum of the other two:
:$\set {1, 2, 3}$ | We note that if $\set {a, b, c}$ is such a set, then $\set {k a, k b, k c}$ satisfy the same properties trivially.
Hence the specification that $\set {a, b, c}$ is a coprime set.
We have that:
:$5 \times 1 = 2 + 3$ so $1 \divides 2 + 3$
:$2 \times 2 = 1 + 3$ so $2 \divides 1 + 3$
:$1 \times 3 = 1 + 2$ so $3 \divides 1 ... | There exists [[Definition:Unique|exactly one]] [[Definition:Set|set]] of [[Definition:Distinct Elements|distinct]] [[Definition:Coprime Integers|coprime]] [[Definition:Positive Integer|positive integers]] such that each is a [[Definition:Divisor of Integer|divisor]] of the [[Definition:Integer Addition|sum]] of the oth... | We note that if $\set {a, b, c}$ is such a [[Definition:Set|set]], then $\set {k a, k b, k c}$ satisfy the same properties trivially.
Hence the specification that $\set {a, b, c}$ is a [[Definition:Coprime Integers|coprime set]].
We have that:
:$5 \times 1 = 2 + 3$ so $1 \divides 2 + 3$
:$2 \times 2 = 1 + 3$ so $2 ... | Set of 3 Integers each Divisor of Sum of Other Two | https://proofwiki.org/wiki/Set_of_3_Integers_each_Divisor_of_Sum_of_Other_Two | https://proofwiki.org/wiki/Set_of_3_Integers_each_Divisor_of_Sum_of_Other_Two | [
"Divisors"
] | [
"Definition:Unique",
"Definition:Set",
"Definition:Distinct/Plural",
"Definition:Coprime/Integers",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers"
] | [
"Definition:Set",
"Definition:Coprime/Integers",
"Definition:Set",
"Definition:Odd Integer",
"Euclid's Lemma",
"Absolute Value of Integer is not less than Divisors",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Euclid's Lemma",
"Ab... |
proofwiki-12299 | Only Number which is Sum of 3 Factors is 6 | The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors is $6$. | Let $n$ be such a positive integer with corresponding divisors $a, b, c$ such that:
:$a + b + c = n$
We note that the set $\set {k a, k b, k c}$ satisfy the same properties trivially as divisors of $k n$.
Hence the specification that $\set {a, b, c}$ is a coprime set.
{{WLOG}}, suppose $a < b < c$.
Since $a, b, c$ are ... | The only [[Definition:Positive Integer|positive integer]] which is the sum of exactly $3$ of its [[Definition:Distinct Elements|distinct]] [[Definition:Coprime Integers|coprime]] [[Definition:Divisor of Integer|divisors]] is $6$. | Let $n$ be such a [[Definition:Positive Integer|positive integer]] with corresponding [[Definition:Divisor of Integer|divisors]] $a, b, c$ such that:
:$a + b + c = n$
We note that the set $\set {k a, k b, k c}$ satisfy the same properties trivially as [[Definition:Divisor of Integer|divisors]] of $k n$.
Hence the spe... | Only Number which is Sum of 3 Factors is 6 | https://proofwiki.org/wiki/Only_Number_which_is_Sum_of_3_Factors_is_6 | https://proofwiki.org/wiki/Only_Number_which_is_Sum_of_3_Factors_is_6 | [
"6",
"Divisors"
] | [
"Definition:Positive/Integer",
"Definition:Distinct/Plural",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers",
"Definition:Strictly Positive/Integer",
"Definition:Contradiction",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Euclid's Lemma",
"Abs... |
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