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-13700 | Pandigital Numbers Divisible by All Integers up to 18 | The following pandigital integers are divisible by all the positive integers up to $18$:
:$2 \, 438 \, 195 \, 760$
:$3 \, 785 \, 942 \, 160$
:$4 \, 753 \, 869 \, 120$
:$4 \, 876 \, 391 \, 520$ | {{begin-eqn}}
{{eqn | l = 2 \, 438 \, 195 \, 760
| r = 2 \times 1 \, 219 \, 097 \, 880
}}
{{eqn | r = 3 \times 812 \, 731 \, 920}}
{{eqn | r = 4 \times 609 \, 548 \, 940}}
{{eqn | r = 5 \times 487 \, 639 \, 152}}
{{eqn | r = 6 \times 406 \, 365 \, 960}}
{{eqn | r = 7 \times 348 \, 313 \, 680}}
{{eqn | r = 8 \time... | The following [[Definition:Pandigital Integer|pandigital integers]] are [[Definition:Divisor of Integer|divisible]] by all the [[Definition:Positive Integer|positive integers]] up to $18$:
:$2 \, 438 \, 195 \, 760$
:$3 \, 785 \, 942 \, 160$
:$4 \, 753 \, 869 \, 120$
:$4 \, 876 \, 391 \, 520$ | {{begin-eqn}}
{{eqn | l = 2 \, 438 \, 195 \, 760
| r = 2 \times 1 \, 219 \, 097 \, 880
}}
{{eqn | r = 3 \times 812 \, 731 \, 920}}
{{eqn | r = 4 \times 609 \, 548 \, 940}}
{{eqn | r = 5 \times 487 \, 639 \, 152}}
{{eqn | r = 6 \times 406 \, 365 \, 960}}
{{eqn | r = 7 \times 348 \, 313 \, 680}}
{{eqn | r = 8 \time... | Pandigital Numbers Divisible by All Integers up to 18 | https://proofwiki.org/wiki/Pandigital_Numbers_Divisible_by_All_Integers_up_to_18 | https://proofwiki.org/wiki/Pandigital_Numbers_Divisible_by_All_Integers_up_to_18 | [
"Pandigital Integers",
"Pandigital Numbers Divisible by All Integers up to 18"
] | [
"Definition:Pandigital Set/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Positive/Integer"
] | [] |
proofwiki-13701 | Smallest n for which 2^n-3 is Divisible by n | The smallest positive integer $n$ such that $2^n - 3$ is divisible by $n$ is $4 \, 700 \, 063 \, 497$. | {{ProofWanted|Brute force by the Lehmers, probably}} | The smallest [[Definition:Positive Integer|positive integer]] $n$ such that $2^n - 3$ is [[Definition:Divisor of Integer|divisible]] by $n$ is $4 \, 700 \, 063 \, 497$. | {{ProofWanted|Brute force by the Lehmers, probably}} | Smallest n for which 2^n-3 is Divisible by n | https://proofwiki.org/wiki/Smallest_n_for_which_2^n-3_is_Divisible_by_n | https://proofwiki.org/wiki/Smallest_n_for_which_2^n-3_is_Divisible_by_n | [
"Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-13702 | Smallest Odd Abundant Number not Divisible by 3 | The smallest odd abundant number not divisible by $3$ is $5 \, 391 \, 411 \, 025$. | We have:
:$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
showing it is not divisible by $3$.
Then from {{DSFLink|5,391,411,025|5 \, 391 \, 411 \, 025}} we have:
:$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 = 2 \times 5 \, 391 \, 411 \, 025 + 1... | The smallest [[Definition:Odd Integer|odd]] [[Definition:Abundant Number|abundant number]] not [[Definition:Divisor of Integer|divisible]] by $3$ is $5 \, 391 \, 411 \, 025$. | We have:
:$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
showing it is not [[Definition:Divisor of Integer|divisible]] by $3$.
Then from {{DSFLink|5,391,411,025|5 \, 391 \, 411 \, 025}} we have:
:$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 =... | Smallest Odd Abundant Number not Divisible by 3 | https://proofwiki.org/wiki/Smallest_Odd_Abundant_Number_not_Divisible_by_3 | https://proofwiki.org/wiki/Smallest_Odd_Abundant_Number_not_Divisible_by_3 | [
"Abundant Numbers",
"5,391,411,025"
] | [
"Definition:Odd Integer",
"Definition:Abundant Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Abundant Number"
] |
proofwiki-13703 | Closure of Irreducible Subspace is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $Y \subseteq S$ be a subset of $S$ which is irreducible in $T$.
Then its closure $Y^-$ in $T$ is also irreducible in $T$. | By definition, $Y$ is an irreducible subset of $S$ in $T$ {{iff}} the subspace $\struct {Y, \tau_Y}$ is an irreducible topological space.
That is, such that two arbitrary non-empty open sets of $\struct {Y, \tau_Y}$ are not disjoint.
The open sets of $T$ in $Y^-$ are the same as the open sets of $\struct {Y, \tau_Y}$.
... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $Y \subseteq S$ be a [[Definition:Subset|subset]] of $S$ which is [[Definition:Irreducible Subset|irreducible]] in $T$.
Then its [[Definition:Closure (Topology)|closure]] $Y^-$ in $T$ is also [[Definition:Irreducible Subset|irre... | By definition, $Y$ is an [[Definition:Irreducible Subset|irreducible subset]] of $S$ in $T$ {{iff}} the [[Definition:Topological Subspace|subspace]] $\struct {Y, \tau_Y}$ is an [[Definition:Irreducible Space|irreducible topological space.]]
That is, such that two arbitrary [[Definition:Non-Empty Set|non-empty]] [[Defi... | Closure of Irreducible Subspace is Irreducible/Proof 1 | https://proofwiki.org/wiki/Closure_of_Irreducible_Subspace_is_Irreducible | https://proofwiki.org/wiki/Closure_of_Irreducible_Subspace_is_Irreducible/Proof_1 | [
"Closure of Irreducible Subspace is Irreducible",
"Irreducible Spaces",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Irreducible Subset",
"Definition:Closure (Topology)",
"Definition:Irreducible Subset"
] | [
"Definition:Irreducible Subset",
"Definition:Topological Subspace",
"Definition:Irreducible Space",
"Definition:Non-Empty Set",
"Definition:Open Set",
"Definition:Disjoint Sets",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Irreducible Space",
"Definition:Open Set",
"Definition:Disj... |
proofwiki-13704 | Closure of Irreducible Subspace is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $Y \subseteq S$ be a subset of $S$ which is irreducible in $T$.
Then its closure $Y^-$ in $T$ is also irreducible in $T$. | In view of {{Defof|Irreducible Subset}}, it suffices to show that for all closed sets $A_1$ and $A_2$ in $T$:
:$Y^- \subseteq A_1 \cup A_2 \implies \exists i_0 \in \set {1, 2} : Y^- \subseteq A_{i_0}$
To this end, let $A_1$ and $A_2$ be closed sets in $T$ such that:
:$Y^- \subseteq A_1 \cup A_2$
Then, in particular:
:$... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $Y \subseteq S$ be a [[Definition:Subset|subset]] of $S$ which is [[Definition:Irreducible Subset|irreducible]] in $T$.
Then its [[Definition:Closure (Topology)|closure]] $Y^-$ in $T$ is also [[Definition:Irreducible Subset|irre... | In view of {{Defof|Irreducible Subset}}, it suffices to show that for all [[Definition:Closed Set (Topology)|closed sets]] $A_1$ and $A_2$ in $T$:
:$Y^- \subseteq A_1 \cup A_2 \implies \exists i_0 \in \set {1, 2} : Y^- \subseteq A_{i_0}$
To this end, let $A_1$ and $A_2$ be [[Definition:Closed Set (Topology)|closed se... | Closure of Irreducible Subspace is Irreducible/Proof 2 | https://proofwiki.org/wiki/Closure_of_Irreducible_Subspace_is_Irreducible | https://proofwiki.org/wiki/Closure_of_Irreducible_Subspace_is_Irreducible/Proof_2 | [
"Closure of Irreducible Subspace is Irreducible",
"Irreducible Spaces",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Irreducible Subset",
"Definition:Closure (Topology)",
"Definition:Irreducible Subset"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Irreducible Subset",
"Closure of Subset of Closed Set of Topological Space is Subset"
] |
proofwiki-13705 | Point is Contained in Irreducible Component | Let $X$ be a topological space.
Let $x\in X$ be a point.
Then $x$ is contained in some irreducible component of $X$. | Because Trivial Topological Space is Irreducible, this is a special case of Irreducible Subspace is Contained in Irreducible Component.
{{qed}}
Category:Irreducible Spaces
0c7d4q4hjsg3o0hxvk280dbnknp8qme | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $x\in X$ be a point.
Then $x$ is [[Definition:Set Inclusion|contained]] in some [[Definition:Irreducible Component|irreducible component]] of $X$. | Because [[Trivial Topological Space is Irreducible]], this is a special case of [[Irreducible Subspace is Contained in Irreducible Component]].
{{qed}}
[[Category:Irreducible Spaces]]
0c7d4q4hjsg3o0hxvk280dbnknp8qme | Point is Contained in Irreducible Component | https://proofwiki.org/wiki/Point_is_Contained_in_Irreducible_Component | https://proofwiki.org/wiki/Point_is_Contained_in_Irreducible_Component | [
"Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Irreducible Component"
] | [
"Trivial Topological Space is Irreducible",
"Irreducible Subspace is Contained in Irreducible Component",
"Category:Irreducible Spaces"
] |
proofwiki-13706 | Irreducible Subspace is Contained in Irreducible Component | Let $T = \struct {S_1, \tau}$ be a non-empty topological space.
Let $S_2$ be an irreducible subset of $S_1$.
Then there exists an irreducible component $S_3$ of $T$ such that $S_2 \subseteq S_3$. | By definition, an irreducible component of $X$ is an irreducible subspace that is maximal among the irreducible subspaces, ordered by the subset relation.
Consider the ordered set $\struct {\AA, \subseteq}$, where:
:$\AA := \leftset {S : S}$ is an irreducible subset of $S_1$ such that $\rightset {S_2 \subseteq S}$
We n... | Let $T = \struct {S_1, \tau}$ be a non-[[Definition:Empty Topological Space|empty]] [[Definition:Topological Space|topological space]].
Let $S_2$ be an [[Definition:Irreducible Subset|irreducible subset]] of $S_1$.
Then there exists an [[Definition:Irreducible Component|irreducible component]] $S_3$ of $T$ such that... | By definition, an [[Definition:Irreducible Component|irreducible component]] of $X$ is an [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]] that is [[Definition:Maximal Element|maximal]] among the [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subs... | Irreducible Subspace is Contained in Irreducible Component | https://proofwiki.org/wiki/Irreducible_Subspace_is_Contained_in_Irreducible_Component | https://proofwiki.org/wiki/Irreducible_Subspace_is_Contained_in_Irreducible_Component | [
"Irreducible Spaces"
] | [
"Definition:Empty Topological Space",
"Definition:Topological Space",
"Definition:Irreducible Subset",
"Definition:Irreducible Component"
] | [
"Definition:Irreducible Component",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Maximal/Element",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Set Ordered by Subset Relation",
"Definition:Ordered Set",
"Definition:Irreducible Subse... |
proofwiki-13707 | Irreducible Component is Closed | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $Y$ be an irreducible component of $T$.
Then $Y$ is closed in $T$. | By Closure of Irreducible Subspace is Irreducible, the closure $Y^-$ of $Y$ is irreducible.
By Set is Subset of its Topological Closure, $Y \subseteq Y^-$.
Because $Y$ is an irreducible component, we must have $Y = Y^-$.
By Set is Closed iff Equals Topological Closure, $Y$ is closed in $T$.
{{qed}}
Category:Irreducibl... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $Y$ be an [[Definition:Irreducible Component|irreducible component]] of $T$.
Then $Y$ is [[Definition:Closed Set (Topology)|closed]] in $T$. | By [[Closure of Irreducible Subspace is Irreducible]], the [[Definition:Topological Closure|closure]] $Y^-$ of $Y$ is [[Definition:Irreducible Space|irreducible]].
By [[Set is Subset of its Topological Closure]], $Y \subseteq Y^-$.
Because $Y$ is an [[Definition:Irreducible Component|irreducible component]], we must ... | Irreducible Component is Closed | https://proofwiki.org/wiki/Irreducible_Component_is_Closed | https://proofwiki.org/wiki/Irreducible_Component_is_Closed | [
"Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Irreducible Component",
"Definition:Closed Set/Topology"
] | [
"Closure of Irreducible Subspace is Irreducible",
"Definition:Closure (Topology)",
"Definition:Irreducible Space",
"Set is Subset of its Topological Closure",
"Definition:Irreducible Component",
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set/Topology",
"Category:Irreducible Spa... |
proofwiki-13708 | Trivial Topological Space is Irreducible | Let $T = \struct {\set s, \tau}$ be a trivial topological space.
Then $T$ is irreducible. | Follows from:
:Trivial Topological Space is Indiscrete
:Indiscrete Space is Irreducible
{{qed}}
Category:Trivial Topological Spaces
Category:Examples of Irreducible Spaces
3a8y3rju7miq9v5j6w9uyrlsu1keece | Let $T = \struct {\set s, \tau}$ be a [[Definition:Trivial Topological Space|trivial topological space]].
Then $T$ is [[Definition:Irreducible Space|irreducible]]. | Follows from:
:[[Trivial Topological Space is Indiscrete]]
:[[Indiscrete Space is Irreducible]]
{{qed}}
[[Category:Trivial Topological Spaces]]
[[Category:Examples of Irreducible Spaces]]
3a8y3rju7miq9v5j6w9uyrlsu1keece | Trivial Topological Space is Irreducible | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Irreducible | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Irreducible | [
"Trivial Topological Spaces",
"Examples of Irreducible Spaces"
] | [
"Definition:Trivial Topological Space",
"Definition:Irreducible Space"
] | [
"Trivial Topological Space is Indiscrete",
"Indiscrete Space is Irreducible",
"Category:Trivial Topological Spaces",
"Category:Examples of Irreducible Spaces"
] |
proofwiki-13709 | Irreducible Components of Hausdorff Space are Points | Let $T = \struct {S, \tau}$ be a non-empty Hausdorff space.
Then the irreducible components of $T$ are the singleton sets. | By Subspace of Hausdorff Space is Hausdorff, the irreducible components of $T$ are also Hausdorff.
By Irreducible Hausdorff Space is Singleton, they can only be singletons.
By Trivial Topological Space is Irreducible, every singleton of $X$ is indeed irreducible.
{{qed}}
Category:Irreducible Spaces
i1l3tsylbrye0j3zoedy... | Let $T = \struct {S, \tau}$ be a non-[[Definition:Empty Topological Space|empty]] [[Definition:Hausdorff Space|Hausdorff space]].
Then the [[Definition:Irreducible Component|irreducible components]] of $T$ are the [[Definition:Singleton|singleton sets]]. | By [[Subspace of Hausdorff Space is Hausdorff]], the [[Definition:Irreducible Component|irreducible components]] of $T$ are also [[Definition:Hausdorff Space|Hausdorff]].
By [[Irreducible Hausdorff Space is Singleton]], they can only be [[Definition:Singleton|singletons]].
By [[Trivial Topological Space is Irreducibl... | Irreducible Components of Hausdorff Space are Points | https://proofwiki.org/wiki/Irreducible_Components_of_Hausdorff_Space_are_Points | https://proofwiki.org/wiki/Irreducible_Components_of_Hausdorff_Space_are_Points | [
"Irreducible Spaces"
] | [
"Definition:Empty Topological Space",
"Definition:T2 Space",
"Definition:Irreducible Component",
"Definition:Singleton"
] | [
"T2 Property is Hereditary",
"Definition:Irreducible Component",
"Definition:T2 Space",
"Irreducible Hausdorff Space is Singleton",
"Definition:Singleton",
"Trivial Topological Space is Irreducible",
"Definition:Singleton",
"Definition:Irreducible Space",
"Category:Irreducible Spaces"
] |
proofwiki-13710 | Pandigital Integer Formed by Digits in Alphabetical Order | The number $8 \, 549 \, 176 \, 320$ is the pandigital integer formed from the digits from $0$ to $9$ arranged in alphabetical order. | {{begin-eqn}}
{{eqn | o =
| r = 8
| c = eight
}}
{{eqn | o =
| r = 5
| c = five
}}
{{eqn | o =
| r = 4
| c = four
}}
{{eqn | o =
| r = 9
| c = nine
}}
{{eqn | o =
| r = 1
| c = one
}}
{{eqn | o =
| r = 7
| c = seven
}}
{{eqn | o =
| r ... | The number $8 \, 549 \, 176 \, 320$ is the [[Definition:Pandigital Integer|pandigital integer]] formed from the [[Definition:Digit|digits]] from $0$ to $9$ arranged in alphabetical order. | {{begin-eqn}}
{{eqn | o =
| r = 8
| c = eight
}}
{{eqn | o =
| r = 5
| c = five
}}
{{eqn | o =
| r = 4
| c = four
}}
{{eqn | o =
| r = 9
| c = nine
}}
{{eqn | o =
| r = 1
| c = one
}}
{{eqn | o =
| r = 7
| c = seven
}}
{{eqn | o =
| r ... | Pandigital Integer Formed by Digits in Alphabetical Order | https://proofwiki.org/wiki/Pandigital_Integer_Formed_by_Digits_in_Alphabetical_Order | https://proofwiki.org/wiki/Pandigital_Integer_Formed_by_Digits_in_Alphabetical_Order | [
"Pandigital Integers",
"8,549,176,320"
] | [
"Definition:Pandigital Set/Integer",
"Definition:Digit"
] | [] |
proofwiki-13711 | Largest Pandigital Square | The largest pandigital square (in the sense where pandigital includes the zero) is $9 \, 814 \, 072 \, 356$:
:$9 \, 814 \, 072 \, 356 = 99 \, 066^2$ | We check all the squares of numbers from $99 \, 067$ up to $\floor {\sqrt {9 \, 876 \, 543 \, 210} } = 99 \, 380$, with the following constraints:
Since all these squares has $9$ as its leftmost digit, the number cannot end with $3$ or $7$.
The number cannot end with $0$ since its square will end in $00$.
A pandigital ... | The largest [[Definition:Pandigital Integer|pandigital]] [[Definition:Square Number|square]] (in the sense where [[Definition:Pandigital Integer|pandigital]] includes the [[Definition:Zero Digit|zero]]) is $9 \, 814 \, 072 \, 356$:
:$9 \, 814 \, 072 \, 356 = 99 \, 066^2$ | We check all the [[Definition:Square Number|squares]] of numbers from $99 \, 067$ up to $\floor {\sqrt {9 \, 876 \, 543 \, 210} } = 99 \, 380$, with the following constraints:
Since all these [[Definition:Square Number|squares]] has $9$ as its leftmost [[Definition:Digit|digit]], the number cannot end with $3$ or $7$... | Largest Pandigital Square | https://proofwiki.org/wiki/Largest_Pandigital_Square | https://proofwiki.org/wiki/Largest_Pandigital_Square | [
"Square Numbers",
"Pandigital Integers",
"9,814,072,356"
] | [
"Definition:Pandigital Set/Integer",
"Definition:Square Number",
"Definition:Pandigital Set/Integer",
"Definition:Zero Digit"
] | [
"Definition:Square Number",
"Definition:Square Number",
"Definition:Digit",
"Definition:Square Number",
"Definition:Pandigital Set/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Pandigital Set/Integer"
] |
proofwiki-13712 | Palindromic Cube with Non-Palindromic Root | The only known palindromic cube with a root that is not itself palindromic is $10 \, 662 \, 526 \, 601$. | We have that:
:$10 \, 662 \, 526 \, 601 = 2201^3$
There are no others whose cube root is below $10^{15}$. | The only known [[Definition:Palindromic Integer|palindromic]] [[Definition:Cube Number|cube]] with a [[Definition:Cube Root|root]] that is not itself [[Definition:Palindromic Integer|palindromic]] is $10 \, 662 \, 526 \, 601$. | We have that:
:$10 \, 662 \, 526 \, 601 = 2201^3$
There are no others whose [[Definition:Cube Root|cube root]] is below $10^{15}$. | Palindromic Cube with Non-Palindromic Root | https://proofwiki.org/wiki/Palindromic_Cube_with_Non-Palindromic_Root | https://proofwiki.org/wiki/Palindromic_Cube_with_Non-Palindromic_Root | [
"Cube Numbers",
"Palindromic Numbers",
"10,662,526,601"
] | [
"Definition:Palindromic Number",
"Definition:Cube Number",
"Definition:Cube Root",
"Definition:Palindromic Number"
] | [
"Definition:Cube Root"
] |
proofwiki-13713 | Open Set of Irreducible Space is Irreducible | Let $T = \struct {S, \tau}$ be an irreducible topological space.
Let $U$ be a non-empty open set of $T$.
Then $U$ is irreducible in its induced subspace topology. | Let $T = \struct {S, \tau}$ be an irreducible topological space.
Let $U$ be a non-empty open set of $T$.
{{AimForCont}} $U$ is not irreducible in $T$.
Then $U = V_1 \cup V_2$ for some closed sets $V_1$ and $V_2$ of $\struct {U, \tau_U}$.
By definition of subspace topology:
:$V_1 = U \cap W_1$
and:
:$V_2 = U \cap W_2$
f... | Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
Let $U$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open set]] of $T$.
Then $U$ is [[Definition:Irreducible Space|irreducible]] in its [[Definition:Subspace Topology|induced subspace top... | Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
Let $U$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open set]] of $T$.
{{AimForCont}} $U$ is not [[Definition:Irreducible Space|irreducible]] in $T$.
Then $U = V_1 \cup V_2$ for some [[D... | Open Set of Irreducible Space is Irreducible | https://proofwiki.org/wiki/Open_Set_of_Irreducible_Space_is_Irreducible | https://proofwiki.org/wiki/Open_Set_of_Irreducible_Space_is_Irreducible | [
"Irreducible Spaces"
] | [
"Definition:Irreducible Space",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace"
] | [
"Definition:Irreducible Space",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Irreducible Space",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Contradiction",
"Definition:Proper Subset",
"Definition:Pro... |
proofwiki-13714 | Union of Open Irreducible Non-Disjoint Subspaces is Irreducible | Let $T = \struct {S, \tau}$ be an irreducible topological space.
Let $U$ and $V$ be open irreducible subspaces of $T$.
Let their intersection $U \cap V$ be non-empty.
Then their union $U \cup V$ is an irreducible subspace of $T$. | In view of the definition of the subspace topology $\tau_{U \cup V}$, it suffices to show that for all closed sets $A_1$ and $A_2$:
:$U \cup V \subseteq A_1 \cup A_2 \implies \exists i_0 \in \set {1, 2} : U \cup V \subseteq A_{i_0}$
To this end, let $A_1$ and $A_2$ closed sets such that:
:$U \cup V \subseteq A_1 \cup A... | Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
Let $U$ and $V$ be [[Definition:Open Set (Topology)|open]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspaces]] of $T$.
Let their [[Definition:Set Intersection|intersection]] $U \cap... | In view of the definition of the [[Definition:Topological Subspace|subspace topology]] $\tau_{U \cup V}$, it suffices to show that for all [[Definition:Closed Set (Topology)|closed sets]] $A_1$ and $A_2$:
:$U \cup V \subseteq A_1 \cup A_2 \implies \exists i_0 \in \set {1, 2} : U \cup V \subseteq A_{i_0}$
To this end,... | Union of Open Irreducible Non-Disjoint Subspaces is Irreducible | https://proofwiki.org/wiki/Union_of_Open_Irreducible_Non-Disjoint_Subspaces_is_Irreducible | https://proofwiki.org/wiki/Union_of_Open_Irreducible_Non-Disjoint_Subspaces_is_Irreducible | [
"Irreducible Spaces"
] | [
"Definition:Irreducible Space",
"Definition:Open Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Definition:Set Union",
"Definition:Irreducible Space",
"Definition:Topological Subspace"
] | [
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"Definition:Non-Empty Set",
"Definition:Proper Subset",
"Definition:Irreducible Space",
"Defin... |
proofwiki-13715 | Closed Set of Ultraconnected Space is Ultraconnected | Let $T = \struct {S, \tau}$ be an ultraconnected topological space.
Let $F \subset S$ be a closed set in $T$.
Then $F$ is ultraconnected. | Let $A, B$ be two non-empty closed sets in $\struct {F, \tau}$.
By Closed Set in Topological Subspace, $A, B$ are closed in $T$ as well.
By {{Defof|Ultraconnected Space}}, $A$ and $B$ are not disjoint.
Since $A$, $B$ are arbitrary, no two non-empty closed sets of $\struct {F, \tau}$ are disjoint.
Hence the result from ... | Let $T = \struct {S, \tau}$ be an [[Definition:Ultraconnected Space|ultraconnected topological space]].
Let $F \subset S$ be a [[Definition:Closed Set (Topology)|closed set]] in $T$.
Then $F$ is [[Definition:Ultraconnected Space|ultraconnected]]. | Let $A, B$ be two [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set (Topology)|closed sets]] in $\struct {F, \tau}$.
By [[Closed Set in Topological Subspace/Corollary|Closed Set in Topological Subspace]], $A, B$ are [[Definition:Closed Set (Topology)|closed]] in $T$ as well.
By {{Defof|Ultraconnected Spa... | Closed Set of Ultraconnected Space is Ultraconnected | https://proofwiki.org/wiki/Closed_Set_of_Ultraconnected_Space_is_Ultraconnected | https://proofwiki.org/wiki/Closed_Set_of_Ultraconnected_Space_is_Ultraconnected | [
"Ultraconnected Spaces"
] | [
"Definition:Ultraconnected Space",
"Definition:Closed Set/Topology",
"Definition:Ultraconnected Space"
] | [
"Definition:Non-Empty Set",
"Definition:Closed Set/Topology",
"Closed Set in Topological Subspace/Corollary",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Non-Empty Set",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets"
] |
proofwiki-13716 | Noetherian Topological Space is Compact | Let $T = \struct {X, \tau}$ be a Noetherian topological space.
Then $T$ is compact. | Let $\family {U_i}_{i \mathop \in I}$ be a cover of $X$.
That is:
:$\ds \bigcup_{i \mathop \in I} U_i = X$
Let $V$ be the collection of finite cover of $\family {U_i}_{i \mathop \in I}$.
{{explain|Exactly what is meant by the above? Does it mean "the set / collection of all finite covers of $X$"? If not, can it be expl... | Let $T = \struct {X, \tau}$ be a [[Definition:Noetherian Topological Space|Noetherian topological space]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | Let $\family {U_i}_{i \mathop \in I}$ be a [[Definition:Cover of Set|cover]] of $X$.
That is:
:$\ds \bigcup_{i \mathop \in I} U_i = X$
Let $V$ be the collection of [[Definition:Finite Cover|finite cover]] of $\family {U_i}_{i \mathop \in I}$.
{{explain|Exactly what is meant by the above? Does it mean "the set / coll... | Noetherian Topological Space is Compact/Proof 1 | https://proofwiki.org/wiki/Noetherian_Topological_Space_is_Compact | https://proofwiki.org/wiki/Noetherian_Topological_Space_is_Compact/Proof_1 | [
"Noetherian Topological Space is Compact",
"Noetherian Topological Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Noetherian Topological Space",
"Definition:Compact Topological Space"
] | [
"Definition:Cover of Set",
"Definition:Cover of Set/Finite",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Maximal/Element",
"Definition:Set Ordered by Subset Relation",
"Definition:Maximal/Element",
"Definition:Neighborhood (Topology)/Point",
"Definition:Contradiction",
"Definitio... |
proofwiki-13717 | Noetherian Topological Space is Compact | Let $T = \struct {X, \tau}$ be a Noetherian topological space.
Then $T$ is compact. | {{Recall|Compact Topological Space|index = 1|compact topological space}}
{{:Definition:Compact Topological Space/Definition 1}}
We may assume $X \ne \O$, since the claim is otherwise trivial.
Let $\CC \subseteq \tau$ be an arbitrary cover for $X$.
We shall show that $\CC$ has a finite subcover.
Consider:
:$A := \leftse... | Let $T = \struct {X, \tau}$ be a [[Definition:Noetherian Topological Space|Noetherian topological space]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | {{Recall|Compact Topological Space|index = 1|compact topological space}}
{{:Definition:Compact Topological Space/Definition 1}}
We may assume $X \ne \O$, since the claim is otherwise trivial.
Let $\CC \subseteq \tau$ be an arbitrary [[Definition:Cover of Set|cover]] for $X$.
We shall show that $\CC$ has a [[Definit... | Noetherian Topological Space is Compact/Proof 2 | https://proofwiki.org/wiki/Noetherian_Topological_Space_is_Compact | https://proofwiki.org/wiki/Noetherian_Topological_Space_is_Compact/Proof_2 | [
"Noetherian Topological Space is Compact",
"Noetherian Topological Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Noetherian Topological Space",
"Definition:Compact Topological Space"
] | [
"Definition:Cover of Set",
"Definition:Subcover/Finite",
"Definition:Finite Subset",
"Definition:Maximal/Element",
"Definition:Cover of Set",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Subcover/Finite"
] |
proofwiki-13718 | Subspace of Noetherian Topological Space is Noetherian | Let $X$ be a Noetherian topological space.
Let $Y \subseteq X$ be a subspace.
Then $Y$ is Noetherian. | Let $Y_1 \subset Y_2 \subset \ldots \subset$ be an ascending chain of open sets in $Y$.
{{explain|If we are going to use the term "ascending chain", then we need to have a page that defines exactly that.}}
By definition of subspace topology, there exists open sets $X_1, X_2, \ldots$ such that
:$X_i \cap Y = Y_i$
for al... | Let $X$ be a [[Definition:Noetherian Topological Space|Noetherian topological space]].
Let $Y \subseteq X$ be a [[Definition:Topological Subspace|subspace]].
Then $Y$ is [[Definition:Noetherian Topological Space|Noetherian]]. | Let $Y_1 \subset Y_2 \subset \ldots \subset$ be an [[Definition:Ascending Chain|ascending chain]] of [[Definition:Open Set (Topology)|open sets]] in $Y$.
{{explain|If we are going to use the term "ascending chain", then we need to have a page that defines exactly that.}}
By definition of [[Definition:Topological Subs... | Subspace of Noetherian Topological Space is Noetherian | https://proofwiki.org/wiki/Subspace_of_Noetherian_Topological_Space_is_Noetherian | https://proofwiki.org/wiki/Subspace_of_Noetherian_Topological_Space_is_Noetherian | [
"Noetherian Topological Spaces"
] | [
"Definition:Noetherian Topological Space",
"Definition:Topological Subspace",
"Definition:Noetherian Topological Space"
] | [
"Definition:Ascending Chain",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Set Intersection",
"Definition:Noetherian Topological Space",
"Definition:Ascending Chain",
"Definition:Open Set/Topology",
"Definition:Noetherian Topological Space"
] |
proofwiki-13719 | Zero Locus of Larger Set is Smaller | Let $k$ be a field.
Let $n \ge 1$ be a natural number.
Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.
Let $I, J \subseteq A$ be subsets, and $\map V I$ and $\map V J$ their zero loci.
Let $I \subseteq J$.
Then $\map V I \supseteq \map V J$. | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map V J
}}
{{eqn | ll= \leadsto
| q = \forall f \in J
| l = \map f x
| r = 0
}}
{{eqn | ll= \leadsto
| q = \forall f \in I
| l = \map f x
| r = 0
| c = since $I \subseteq J$ by assumption
}}
{{eqn | ll= \leadsto
| l... | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the [[Definition:Ring of Polynomials|ring of polynomials]] in $n$ [[Definition:Variable of Polynomial Ring|variables]] over $k$.
Let $I, J \subseteq A$... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map V J
}}
{{eqn | ll= \leadsto
| q = \forall f \in J
| l = \map f x
| r = 0
}}
{{eqn | ll= \leadsto
| q = \forall f \in I
| l = \map f x
| r = 0
| c = since $I \subseteq J$ by assumption
}}
{{eqn | ll= \leadsto
| l... | Zero Locus of Larger Set is Smaller | https://proofwiki.org/wiki/Zero_Locus_of_Larger_Set_is_Smaller | https://proofwiki.org/wiki/Zero_Locus_of_Larger_Set_is_Smaller | [
"Algebraic Geometry"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Natural Numbers",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Subset",
"Definition:Zero Locus of Set of Polynomials"
] | [
"Category:Algebraic Geometry"
] |
proofwiki-13720 | Smallest Multiply Perfect Number of Order 5 | The number $14 \, 182 \, 439 \, 040$ is multiply perfect of order $5$:
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200 = 5 \times 14 \, 182 \, 439 \, 040$
It is the smallest positive integer to be so. | From {{DSFLink|14,182,439,040|14 \, 182 \, 439 \, 040}}:
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200$
{{ProofWanted|That it is the smallest one remains to be proved.}} | The number $14 \, 182 \, 439 \, 040$ is [[Definition:Multiply Perfect Number|multiply perfect]] of [[Definition:Order of Multiply Perfect Number|order]] $5$:
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200 = 5 \times 14 \, 182 \, 439 \, 040$
It is the smallest [[Definition:Positive Integer|posi... | From {{DSFLink|14,182,439,040|14 \, 182 \, 439 \, 040}}:
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200$
{{ProofWanted|That it is the smallest one remains to be proved.}} | Smallest Multiply Perfect Number of Order 5 | https://proofwiki.org/wiki/Smallest_Multiply_Perfect_Number_of_Order_5 | https://proofwiki.org/wiki/Smallest_Multiply_Perfect_Number_of_Order_5 | [
"Multiply Perfect Numbers",
"14,182,439,040"
] | [
"Definition:Multiply Perfect Number",
"Definition:Multiply Perfect Number/Order",
"Definition:Positive/Integer"
] | [] |
proofwiki-13721 | Smallest Fourth Power expressible as Sum of 4 Fourth Powers | $15 \, 527 \, 402 \, 881$ is the smallest fourth power which can be expressed as the sum of $4$ fourth powers:
:$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$ | {{begin-eqn}}
{{eqn | o =
| r = 30^4 + 120^4 + 272^4 + 315^4
| c =
}}
{{eqn | r = 810 \, 000 + 207 \, 360 \, 000 + 5 \, 473 \, 632 \, 256 + 9 \, 845 \, 600 \, 625
| c =
}}
{{eqn | r = 15 \, 527 \, 402 \, 881
| c =
}}
{{eqn | r = 353^4
| c =
}}
{{end-eqn}}
{{ProofWanted|That it is the ... | $15 \, 527 \, 402 \, 881$ is the smallest [[Definition:Fourth Power|fourth power]] which can be expressed as the [[Definition:Integer Addition|sum]] of $4$ [[Definition:Fourth Power|fourth powers]]:
:$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$ | {{begin-eqn}}
{{eqn | o =
| r = 30^4 + 120^4 + 272^4 + 315^4
| c =
}}
{{eqn | r = 810 \, 000 + 207 \, 360 \, 000 + 5 \, 473 \, 632 \, 256 + 9 \, 845 \, 600 \, 625
| c =
}}
{{eqn | r = 15 \, 527 \, 402 \, 881
| c =
}}
{{eqn | r = 353^4
| c =
}}
{{end-eqn}}
{{ProofWanted|That it is the... | Smallest Fourth Power expressible as Sum of 4 Fourth Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_expressible_as_Sum_of_4_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Fourth_Power_expressible_as_Sum_of_4_Fourth_Powers | [
"Fourth Powers",
"15,527,402,881"
] | [
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13722 | Largest Known Lead by 4n+1 in Prime Number Race | In the prime number race between prime numbers of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of integers where $4 n + 1$ is not less than $4 n - 1$ is between $18 \, 465 \, 126 \, 293$ and $19 \, 033 \, 524 \, 538$. | First note that:
{{begin-eqn}}
{{eqn | l = 18 \, 465 \, 126 \, 257
| r = 4 \times 4 \, 616 \, 281 \, 564 + 1
| c =
}}
{{eqn | l = 18 \, 465 \, 126 \, 293
| r = 4 \times 4 \, 616 \, 281 \, 573 + 1
| c =
}}
{{end-eqn}}
where it can be seen that prime numbers of the form $4 n + 1$ are locally inc... | In the [[Definition:Prime Number Race|prime number race]] between [[Definition:Prime Number|prime numbers]] of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of [[Definition:Integer|integers]] where $4 n + 1$ is not less than $4 n - 1$ is between $18 \, 465 \, 126 \, 293$ and $19 \, 033 \, 524 \, 538$. | First note that:
{{begin-eqn}}
{{eqn | l = 18 \, 465 \, 126 \, 257
| r = 4 \times 4 \, 616 \, 281 \, 564 + 1
| c =
}}
{{eqn | l = 18 \, 465 \, 126 \, 293
| r = 4 \times 4 \, 616 \, 281 \, 573 + 1
| c =
}}
{{end-eqn}}
where it can be seen that [[Definition:Prime Number|prime numbers]] of the ... | Largest Known Lead by 4n+1 in Prime Number Race | https://proofwiki.org/wiki/Largest_Known_Lead_by_4n+1_in_Prime_Number_Race | https://proofwiki.org/wiki/Largest_Known_Lead_by_4n+1_in_Prime_Number_Race | [
"Prime Number Races"
] | [
"Definition:Prime Number Race",
"Definition:Prime Number",
"Definition:Integer"
] | [
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13723 | Number whose Square is in 2 Identical Halves | The number $36 \, 363 \, 636 \, 364$ has the property that its square can be split into two identical halves:
:$36 \, 363 \, 636 \, 364 = 1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496$ | It just does. | The number $36 \, 363 \, 636 \, 364$ has the property that its [[Definition:Square (Algebra)|square]] can be split into two identical halves:
:$36 \, 363 \, 636 \, 364 = 1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496$ | It just does. | Number whose Square is in 2 Identical Halves | https://proofwiki.org/wiki/Number_whose_Square_is_in_2_Identical_Halves | https://proofwiki.org/wiki/Number_whose_Square_is_in_2_Identical_Halves | [
"Square Numbers",
"36,363,636,364"
] | [
"Definition:Square/Function"
] | [] |
proofwiki-13724 | Fifth Power expressible as Sum of 4 Fifth Powers | $61 \, 917 \, 364 \, 224$ can be expressed as the sum of $4$ fifth powers:
:$61 \, 917 \, 364 \, 224 = 144^5 = 27^5 + 84^5 + 110^5 + 133^5$ | {{begin-eqn}}
{{eqn | o =
| r = 27^5 + 84^5 + 110^5 + 133^5
| c =
}}
{{eqn | r = 14 \, 348 \, 907 + 4 \, 182 \, 119 \, 424 + 16 \, 105 \, 100 \, 000 + 41 \, 615 \, 795 \, 893
| c =
}}
{{eqn | r = 61 \, 917 \, 364 \, 224
| c =
}}
{{eqn | r = 144^5
| c =
}}
{{end-eqn}}
{{qed}} | $61 \, 917 \, 364 \, 224$ can be expressed as the [[Definition:Integer Addition|sum]] of $4$ [[Definition:Fifth Power|fifth powers]]:
:$61 \, 917 \, 364 \, 224 = 144^5 = 27^5 + 84^5 + 110^5 + 133^5$ | {{begin-eqn}}
{{eqn | o =
| r = 27^5 + 84^5 + 110^5 + 133^5
| c =
}}
{{eqn | r = 14 \, 348 \, 907 + 4 \, 182 \, 119 \, 424 + 16 \, 105 \, 100 \, 000 + 41 \, 615 \, 795 \, 893
| c =
}}
{{eqn | r = 61 \, 917 \, 364 \, 224
| c =
}}
{{eqn | r = 144^5
| c =
}}
{{end-eqn}}
{{qed}} | Fifth Power expressible as Sum of 4 Fifth Powers | https://proofwiki.org/wiki/Fifth_Power_expressible_as_Sum_of_4_Fifth_Powers | https://proofwiki.org/wiki/Fifth_Power_expressible_as_Sum_of_4_Fifth_Powers | [
"Fifth Powers",
"61,917,364,224"
] | [
"Definition:Addition/Integers",
"Definition:Fifth Power"
] | [] |
proofwiki-13725 | Common Sum of 3 Distinct Amicable Pairs | The integer $64 \, 795 \, 852 \, 800$ is the sum of $3$ distinct amicable pairs:
:$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$
:$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$
:$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$
all of them odd. | We have that:
From $29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ are amicable:
{{:Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075}}
From $31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$ are amicable:
{{:Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525}}
From $32 \, 129 \, 958 \, 525$ a... | The [[Definition:Integer|integer]] $64 \, 795 \, 852 \, 800$ is the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Distinct|distinct]] [[Definition:Amicable Pair|amicable pairs]]:
:$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$
:$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$
:$32 \, 129 \,... | We have that:
From [[Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075|$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ are amicable]]:
{{:Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075}}
From [[Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525|$31 \, 695 \, 652 \, 275$ and $33 \, 100 ... | Common Sum of 3 Distinct Amicable Pairs | https://proofwiki.org/wiki/Common_Sum_of_3_Distinct_Amicable_Pairs | https://proofwiki.org/wiki/Common_Sum_of_3_Distinct_Amicable_Pairs | [
"Amicable Pairs"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Amicable Pair"
] | [
"Odd Amicable Pair/Examples/29,912,035,725-34,883,817,075",
"Odd Amicable Pair/Examples/31,695,652,275-33,100,200,525",
"Odd Amicable Pair/Examples/32,129,958,525-32,665,894,275"
] |
proofwiki-13726 | Multiply Perfect Number of Order 6 | The number defined as:
:$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
is multiply perfect of o... | From Divisor Sum Function is Multiplicative, we may take each prime factor separately and form $\map {\sigma_1} n$ as the product of the divisor sum of each.
Each of the prime factors which occur with multiplicity $1$ will be treated first.
A prime factor $p$ contributes towards the combined $\sigma_1$ a factor $p + 1$... | The number defined as:
:$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
is [[Definition:Multipl... | From [[Divisor Sum Function is Multiplicative]], we may take each [[Definition:Prime Factor|prime factor]] separately and form $\map {\sigma_1} n$ as the [[Definition:Integer Multiplication|product]] of the [[Definition:Divisor Sum Function|divisor sum]] of each.
Each of the [[Definition:Prime Factor|prime factors]] ... | Multiply Perfect Number of Order 6 | https://proofwiki.org/wiki/Multiply_Perfect_Number_of_Order_6 | https://proofwiki.org/wiki/Multiply_Perfect_Number_of_Order_6 | [
"Multiply Perfect Numbers"
] | [
"Definition:Multiply Perfect Number",
"Definition:Multiply Perfect Number/Order"
] | [
"Divisor Sum Function is Multiplicative",
"Definition:Prime Factor",
"Definition:Multiplication/Integers",
"Definition:Divisor Sum Function",
"Definition:Prime Factor",
"Definition:Prime Decomposition/Multiplicity",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Divisor Sum of P... |
proofwiki-13727 | Probability of Receiving Complete Suit as Hand at Bridge | The probability of being dealt a complete suit in a deal at Bridge is $1$ in $158 \, 753 \, 389 \, 900$. | {{ProofWanted|Straightforward but boring exercise in combinatorics}} | The [[Definition:Probability|probability]] of being dealt a complete [[Definition:Suit of Cards|suit]] in a deal at [[Definition:Bridge (Game)|Bridge]] is $1$ in $158 \, 753 \, 389 \, 900$. | {{ProofWanted|Straightforward but boring exercise in combinatorics}} | Probability of Receiving Complete Suit as Hand at Bridge | https://proofwiki.org/wiki/Probability_of_Receiving_Complete_Suit_as_Hand_at_Bridge | https://proofwiki.org/wiki/Probability_of_Receiving_Complete_Suit_as_Hand_at_Bridge | [
"Bridge (Game)"
] | [
"Definition:Probability",
"Definition:Deck of Cards/Suit",
"Definition:Bridge (Game)"
] | [] |
proofwiki-13728 | Polynomial Ring is Generated by Indeterminate over Ground Ring | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring over $R$.
Let $\iota: R \to R \sqbrk X$ be the embedding.
Then $R \sqbrk X$ is generated by $X$ over $R$. | {{ProofWanted|Use Polynomial is Linear Combination of Monomials}}
Category:Polynomial Theory
2x9lzdnsljh8sd5unxy7zgtw235pu2z | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$.
Let $\iota: R \to R \sqbrk X$ be the [[Definition:Embedding into Polynomial Ring|embedding]].
Then $R \sqbrk X$ is [[Definition:Generator of Ring Extens... | {{ProofWanted|Use [[Polynomial is Linear Combination of Monomials]]}}
[[Category:Polynomial Theory]]
2x9lzdnsljh8sd5unxy7zgtw235pu2z | Polynomial Ring is Generated by Indeterminate over Ground Ring | https://proofwiki.org/wiki/Polynomial_Ring_is_Generated_by_Indeterminate_over_Ground_Ring | https://proofwiki.org/wiki/Polynomial_Ring_is_Generated_by_Indeterminate_over_Ground_Ring | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Embedding",
"Definition:Generator of Ring Extension"
] | [
"Polynomial is Linear Combination of Monomials",
"Category:Polynomial Theory"
] |
proofwiki-13729 | Smallest Cunningham Chain of the First Kind of Length 12 | The smallest Cunningham chain of the first kind of length $12$ is:
:$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$,
::$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 6... | Let $C$ denote the sequence in question.
We have that $554 \, 688 \, 278 \, 429$ is prime.
First note that:
:$\dfrac {554 \, 688 \, 278 \, 429 - 1} 2 = 277 \, 344 \, 139 \, 214 = 2 \times 138 \, 672 \, 069 \, 607$
and so is not prime.
Thus $554 \, 688 \, 278 \, 429$ is not a safe prime, and thus fulfils the requirement... | The smallest [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $12$ is:
:$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$,
::$8 \, 875 \, 012 \, 454 \, 879$, $17 \,... | Let $C$ denote the [[Definition:Sequence|sequence]] in question.
We have that $554 \, 688 \, 278 \, 429$ is [[Definition:Prime Number|prime]].
First note that:
:$\dfrac {554 \, 688 \, 278 \, 429 - 1} 2 = 277 \, 344 \, 139 \, 214 = 2 \times 138 \, 672 \, 069 \, 607$
and so is not [[Definition:Prime Number|prime]].
T... | Smallest Cunningham Chain of the First Kind of Length 12 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_12 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_12 | [
"Cunningham Chains"
] | [
"Definition:Cunningham Chain/First Kind",
"Definition:Length of Sequence"
] | [
"Definition:Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Safe Prime",
"Definition:Cunningham Chain/First Kind",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime... |
proofwiki-13730 | Set of Integers Bounded Above by Real Number has Greatest Element | Let $\Z$ be the set of integers.
Let $\le$ be the usual ordering on the real numbers $\R$.
Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\R, \le}$.
Then $S$ has a greatest element. | Let $S$ be bounded above by $x \in \R$.
By the Axiom of Archimedes, there exists an integer $n \ge x$.
Then $S$ is bounded above by $n$.
By Set of Integers Bounded Above by Integer has Greatest Element, $S$ has a greatest element.
{{qed}} | Let $\Z$ be the [[Definition:Integer|set of integers]].
Let $\le$ be the [[Definition:Usual Ordering|usual ordering on the real numbers]] $\R$.
Let $\O \subset S \subseteq \Z$ such that $S$ is [[Definition:Bounded Above Set|bounded above]] in $\struct {\R, \le}$.
Then $S$ has a [[Definition:Greatest Element|greates... | Let $S$ be [[Definition:Bounded Above Set|bounded above]] by $x \in \R$.
By the [[Axiom of Archimedes]], there exists an [[Definition:Integer|integer]] $n \ge x$.
Then $S$ is [[Definition:Bounded Above Set|bounded above]] by $n$.
By [[Set of Integers Bounded Above by Integer has Greatest Element]], $S$ has a [[Defin... | Set of Integers Bounded Above by Real Number has Greatest Element | https://proofwiki.org/wiki/Set_of_Integers_Bounded_Above_by_Real_Number_has_Greatest_Element | https://proofwiki.org/wiki/Set_of_Integers_Bounded_Above_by_Real_Number_has_Greatest_Element | [
"Set of Integers Bounded Above has Greatest Element"
] | [
"Definition:Integer",
"Definition:Usual Ordering",
"Definition:Bounded Above Set",
"Definition:Greatest Element"
] | [
"Definition:Bounded Above Set",
"Axiom of Archimedes",
"Definition:Integer",
"Definition:Bounded Above Set",
"Set of Integers Bounded Above by Integer has Greatest Element",
"Definition:Greatest Element"
] |
proofwiki-13731 | Set of Integers Bounded Below by Real Number has Smallest Element | Let $\Z$ be the set of integers.
Let $\le$ be the usual ordering on the real numbers $\R$.
Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\R, \le}$.
Then $S$ has a smallest element. | Let $S$ be bounded below by $x \in \R$.
By the Axiom of Archimedes, there exists an integer $n \le x$.
Then $S$ is bounded below by $n$.
By Set of Integers Bounded Below by Integer has Smallest Element, $S$ has a smallest element.
{{qed}} | Let $\Z$ be the [[Definition:Integer|set of integers]].
Let $\le$ be the [[Definition:Usual Ordering|usual ordering on the real numbers]] $\R$.
Let $\O \subset S \subseteq \Z$ such that $S$ is [[Definition:Bounded Below Set|bounded below]] in $\struct {\R, \le}$.
Then $S$ has a [[Definition:Smallest Element|smalles... | Let $S$ be [[Definition:Bounded Below Set|bounded below]] by $x \in \R$.
By the [[Axiom of Archimedes]], there exists an [[Definition:Integer|integer]] $n \le x$.
Then $S$ is [[Definition:Bounded Below Set|bounded below]] by $n$.
By [[Set of Integers Bounded Below by Integer has Smallest Element]], $S$ has a [[Defin... | Set of Integers Bounded Below by Real Number has Smallest Element | https://proofwiki.org/wiki/Set_of_Integers_Bounded_Below_by_Real_Number_has_Smallest_Element | https://proofwiki.org/wiki/Set_of_Integers_Bounded_Below_by_Real_Number_has_Smallest_Element | [
"Set of Integers Bounded Below has Smallest Element"
] | [
"Definition:Integer",
"Definition:Usual Ordering",
"Definition:Bounded Below Set",
"Definition:Smallest Element"
] | [
"Definition:Bounded Below Set",
"Axiom of Archimedes",
"Definition:Integer",
"Definition:Bounded Below Set",
"Set of Integers Bounded Below by Integer has Smallest Element",
"Definition:Smallest Element"
] |
proofwiki-13732 | Real Number lies between Unique Pair of Consecutive Integers | Let $x$ be a real number. | === Existence ===
By Set of Integers Bounded Above by Real Number has Greatest Element, the set:
:$S = \left\{{m \in \Z: m \le x}\right\}$
has a greatest element, say $n$.
Because $n+1>n$, $n+1\notin S$.
Thus $n+1> x$.
Thus $n\leq x < n+1$.
{{qed|lemma}} | Let $x$ be a [[Definition:Real Number|real number]]. | === Existence ===
By [[Set of Integers Bounded Above by Real Number has Greatest Element]], the [[Definition:Set|set]]:
:$S = \left\{{m \in \Z: m \le x}\right\}$
has a [[Definition:Greatest Element|greatest element]], say $n$.
Because $n+1>n$, $n+1\notin S$.
Thus $n+1> x$.
Thus $n\leq x < n+1$.
{{qed|lemma}} | Real Number lies between Unique Pair of Consecutive Integers | https://proofwiki.org/wiki/Real_Number_lies_between_Unique_Pair_of_Consecutive_Integers | https://proofwiki.org/wiki/Real_Number_lies_between_Unique_Pair_of_Consecutive_Integers | [
"Integers",
"Floor Function"
] | [
"Definition:Real Number"
] | [
"Set of Integers Bounded Above by Real Number has Greatest Element",
"Definition:Set",
"Definition:Greatest Element",
"Definition:Greatest Element",
"Definition:Set",
"Definition:Greatest Element"
] |
proofwiki-13733 | Supremum of Set of Integers is Integer | Let $S \subset \Z$ be a non-empty subset of the set of integers.
Let $S$ be bounded above in the set of real numbers.
Then its supremum $\sup S$ is an integer. | By Supremum of Set of Integers equals Greatest Element, $S$ has a greatest element $n \in \Z$, that is equals to the supremum of $S$.
{{qed}} | Let $S \subset \Z$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]].
Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] in the [[Definition:Real Number|set of real numbers]].
Then its [[Definition:Supremum of Subset of Re... | By [[Supremum of Set of Integers equals Greatest Element]], $S$ has a [[Definition:Greatest Element|greatest element]] $n \in \Z$, that is equals to the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
{{qed}} | Supremum of Set of Integers is Integer | https://proofwiki.org/wiki/Supremum_of_Set_of_Integers_is_Integer | https://proofwiki.org/wiki/Supremum_of_Set_of_Integers_is_Integer | [
"Integers"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Integer",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Integer"
] | [
"Supremum of Set of Integers equals Greatest Element",
"Definition:Greatest Element",
"Definition:Supremum of Set/Real Numbers"
] |
proofwiki-13734 | Definition:Constructed Semantics/Instance 1/Rule of Idempotence | The Rule of Idempotence:
:$(p \lor p) \implies p$
is a tautology in Instance 1 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Idempotence can be written as:
: $\neg \left({p \lor p}\right) \lor p$
This evaluates as follows:
:$\begin{array}{|cccc|c|c|} \hline
\neg & (p & \lor & p) & \lor & p \\
\hline
2... | The [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]]:
:$(p \lor p) \implies p$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]]. | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]] can be written as:
: $\n... | Definition:Constructed Semantics/Instance 1/Rule of Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Idempotence | [
"Formal Semantics"
] | [
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 1",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Category:Formal Semantics"
] |
proofwiki-13735 | Definition:Constructed Semantics/Instance 1/Rule of Addition | The Rule of Addition:
:$q \implies (q \lor p)$
is a tautology in Instance 1 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Addition can be written as:
: $\neg q \lor \left({p \lor q}\right)$
This evaluates as follows:
:$\begin{array}{|cc|c|ccc|} \hline
\neg & q & \lor & (p & \lor & q) \\
\hline
2 & ... | The [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]]:
:$q \implies (q \lor p)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|constructed semantics]]. | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]] can be written as:
: $\neg q \lor \left({p... | Definition:Constructed Semantics/Instance 1/Rule of Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Addition | [
"Formal Semantics"
] | [
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 1",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Category:Formal Semantics"
] |
proofwiki-13736 | Definition:Constructed Semantics/Instance 1/Rule of Commutation | The Rule of Commutation:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a tautology in Instance 1 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Commutation can be written as:
:$\neg \left({p \lor q}\right) \lor \left({q \lor p}\right)$
This evaluates as follows:
:$\begin{array}{|cccc|c|ccc|} \hline
\neg & (p & \lor & q)... | The [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]]:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[Definition:Constructed Semantics|cons... | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]] can be written as:
:$\ne... | Definition:Constructed Semantics/Instance 1/Rule of Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Rule_of_Commutation | [
"Formal Semantics"
] | [
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 1",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Category:Formal Semantics"
] |
proofwiki-13737 | Supremum of Set of Integers equals Greatest Element | Let $S \subset \Z$ be a non-empty subset of the set of integers.
Let $S$ be bounded above in the set of real numbers $\R$.
Then $S$ has a greatest element, and it is equal to the supremum $\sup S$. | By Set of Integers Bounded Above by Real Number has Greatest Element, $S$ has a greatest element, say $n \in S$.
By Greatest Element is Supremum, $n$ is the supremum of $S$.
{{qed}} | Let $S \subset \Z$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]].
Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] in the [[Definition:Real Number|set of real numbers]] $\R$.
Then $S$ has a [[Definition:Greatest Elem... | By [[Set of Integers Bounded Above by Real Number has Greatest Element]], $S$ has a [[Definition:Greatest Element|greatest element]], say $n \in S$.
By [[Greatest Element is Supremum]], $n$ is the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
{{qed}} | Supremum of Set of Integers equals Greatest Element | https://proofwiki.org/wiki/Supremum_of_Set_of_Integers_equals_Greatest_Element | https://proofwiki.org/wiki/Supremum_of_Set_of_Integers_equals_Greatest_Element | [
"Integers"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Integer",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Real Number",
"Definition:Greatest Element",
"Definition:Supremum of Set/Real Numbers"
] | [
"Set of Integers Bounded Above by Real Number has Greatest Element",
"Definition:Greatest Element",
"Greatest Element is Supremum",
"Definition:Supremum of Set/Real Numbers"
] |
proofwiki-13738 | Definition:Constructed Semantics/Instance 1/Factor Principle | The Factor Principle:
:$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$
is a tautology in Instance 1 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Factor Principle can be written as:
:$\neg \left({\neg p \lor q}\right) \lor \left({\neg \left({r \lor p}\right) \lor \left ({r \lor q}\right)}\right)$
This evaluates as follows:
:$\beg... | The [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]]:
:$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 1|Instance 1]] of [[D... | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]] can be written as:
:$\neg \left({\neg p \... | Definition:Constructed Semantics/Instance 1/Factor Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Factor_Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_1/Factor_Principle | [
"Formal Semantics"
] | [
"Factor Principles/Disjunction on Left/Formulation 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 1",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Factor Principles/Disjunction on Left/Formulation 2",
"Category:Formal Semantics"
] |
proofwiki-13739 | Weak Inequality of Integers iff Strict Inequality with Integer plus One | Let $a, b \in \Z$ be integers.
{{TFAE}}
:$(1): \quad a \le b$
:$(2): \quad a < b + 1$
where:
:$\le$ is the ordering on the integers
:$<$ is the strict ordering on the integers. | {{ProofWanted}}
Category:Orderings on Integers
o50pjpmex3eprkvlhweb2aux9n6498h | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
{{TFAE}}
:$(1): \quad a \le b$
:$(2): \quad a < b + 1$
where:
:$\le$ is the [[Definition:Ordering on Integers|ordering on the integers]]
:$<$ is the [[Definition:Strict Ordering on Integers|strict ordering on the integers]]. | {{ProofWanted}}
[[Category:Orderings on Integers]]
o50pjpmex3eprkvlhweb2aux9n6498h | Weak Inequality of Integers iff Strict Inequality with Integer plus One | https://proofwiki.org/wiki/Weak_Inequality_of_Integers_iff_Strict_Inequality_with_Integer_plus_One | https://proofwiki.org/wiki/Weak_Inequality_of_Integers_iff_Strict_Inequality_with_Integer_plus_One | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Ordering on Integers",
"Definition:Strict Ordering on Integers"
] | [
"Category:Orderings on Integers"
] |
proofwiki-13740 | Set of Orbits forms Partition | Let $G$ be a group.
Let $X$ be a set.
Let $G$ act on $X$.
Then the set of orbits of the group action forms a partition of $X$. | Follows from the Fundamental Theorem on Equivalence Relations.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $X$ be a [[Definition:Set|set]].
Let $G$ [[Definition:Group Action|act]] on $X$.
Then the [[Definition:Set of Orbits|set of orbits]] of the [[Definition:Group Action|group action]] forms a [[Definition:Set Partition|partition]] of $X$. | Follows from the [[Fundamental Theorem on Equivalence Relations]].
{{qed}} | Set of Orbits forms Partition | https://proofwiki.org/wiki/Set_of_Orbits_forms_Partition | https://proofwiki.org/wiki/Set_of_Orbits_forms_Partition | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action",
"Definition:Orbit (Group Theory)/Set of Orbits",
"Definition:Group Action",
"Definition:Set Partition"
] | [
"Fundamental Theorem on Equivalence Relations"
] |
proofwiki-13741 | Interior of Closure is Regular Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
Then $H^{- \circ}$ is regular open. | We must show that $H^{- \circ - \circ} = H^{- \circ}$.
First we show that $H^{- \circ - \circ} \subseteq H^{- \circ}$.
By the definition of interior:
:$H^{- \circ} \subseteq H^-$
By Topological Closure of Subset is Subset of Topological Closure:
:$H^{- \circ -} \subseteq H^{--} = H^-$
By Interior of Subset:
:$H^{- \cir... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
Then $H^{- \circ}$ is [[Definition:Regular Open Set|regular open]]. | We must show that $H^{- \circ - \circ} = H^{- \circ}$.
First we show that $H^{- \circ - \circ} \subseteq H^{- \circ}$.
By the definition of [[Definition:Interior (Topology)/Definition 2|interior]]:
:$H^{- \circ} \subseteq H^-$
By [[Topological Closure of Subset is Subset of Topological Closure]]:
:$H^{- \circ -} \... | Interior of Closure is Regular Open | https://proofwiki.org/wiki/Interior_of_Closure_is_Regular_Open | https://proofwiki.org/wiki/Interior_of_Closure_is_Regular_Open | [
"Regular Open Sets",
"Set Interiors",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Regular Open Set"
] | [
"Definition:Interior (Topology)/Definition 2",
"Topological Closure of Subset is Subset of Topological Closure",
"Interior of Subset",
"Definition:Interior (Topology)/Definition 2",
"Interior of Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-13742 | Jordan's Lemma | Let $r > 0$ be a real number.
Let:
:$C_r = \set {r e^{i \theta}: 0 \le \theta \le \pi}$
Let $g : C_r \to \C$ be a continuous function.
Define $f : C_r \to \C$ by:
:$\map f z = e^{i a z} \map g z$
for each $z \in C_r$, for some real number $a > 0$.
Then:
:$\ds \size {\int_{C_r} \map f z \rd z} \le \frac \pi a \paren {... | We have:
{{begin-eqn}}
{{eqn | l = \size {\int_{C_r} \map f z \rd z}
| r = \size {\int_{C_r} e^{i a z} \map g z \rd z}
}}
{{eqn | r = \size {i r} \size {\int_0^\pi e^{i a r e^{i \theta} } \map g {r e^{i \theta} } e^{i \theta} \rd \theta}
| c = {{Defof|Complex Contour Integral}}
}}
{{eqn | r = \size {i r} ... | Let $r > 0$ be a [[Definition:Real Number|real number]].
Let:
:$C_r = \set {r e^{i \theta}: 0 \le \theta \le \pi}$
Let $g : C_r \to \C$ be a [[Definition:Continuous Complex Function|continuous function]].
Define $f : C_r \to \C$ by:
:$\map f z = e^{i a z} \map g z$
for each $z \in C_r$, for some [[Definition:Rea... | We have:
{{begin-eqn}}
{{eqn | l = \size {\int_{C_r} \map f z \rd z}
| r = \size {\int_{C_r} e^{i a z} \map g z \rd z}
}}
{{eqn | r = \size {i r} \size {\int_0^\pi e^{i a r e^{i \theta} } \map g {r e^{i \theta} } e^{i \theta} \rd \theta}
| c = {{Defof|Complex Contour Integral}}
}}
{{eqn | r = \size {i r}... | Jordan's Lemma | https://proofwiki.org/wiki/Jordan's_Lemma | https://proofwiki.org/wiki/Jordan's_Lemma | [
"Complex Contour Integrals",
"Complex Analysis"
] | [
"Definition:Real Number",
"Definition:Continuous Complex Function",
"Definition:Real Number"
] | [
"Euler's Formula",
"Triangle Inequality for Integrals/Complex",
"Definite Integral of Constant Multiple of Real Function",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Integration by Substitution",
"Sine of Angle plus Integer Multiple of Pi",
"Sine Function is Odd",
"Jordan's In... |
proofwiki-13743 | Lobachevsky Integral Formula | Let $f$ be a continuous real function, periodic with period $\pi$.
Then:
:$\ds \int_0^\infty \frac {\sin x} x \map f x \rd x = \int_0^{\frac \pi 2} \map f x \rd x$ | {{ProofWanted}}
{{Namedfor|Nikolai Ivanovich Lobachevsky|cat = Lobachevsky}}
Category:Integral Calculus
Category:Definite Integrals involving Sine Function
0zmhrxuk54mssdudgepztl6s6lqsgus | Let $f$ be a [[Definition:Continuous Real Function|continuous real function]], [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $\pi$.
Then:
:$\ds \int_0^\infty \frac {\sin x} x \map f x \rd x = \int_0^{\frac \pi 2} \map f x \rd x$ | {{ProofWanted}}
{{Namedfor|Nikolai Ivanovich Lobachevsky|cat = Lobachevsky}}
[[Category:Integral Calculus]]
[[Category:Definite Integrals involving Sine Function]]
0zmhrxuk54mssdudgepztl6s6lqsgus | Lobachevsky Integral Formula | https://proofwiki.org/wiki/Lobachevsky_Integral_Formula | https://proofwiki.org/wiki/Lobachevsky_Integral_Formula | [
"Integral Calculus",
"Definite Integrals involving Sine Function"
] | [
"Definition:Continuous Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Category:Integral Calculus",
"Category:Definite Integrals involving Sine Function"
] |
proofwiki-13744 | Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series | Let $f, g: \N \to \C$ be arithmetic functions with Dirichlet convolution $h = f * g$.
Let $F, G, H$ be their Dirichlet series.
Let $\sigma_f, \sigma_g, \sigma_h$ be their abscissae of absolute convergence.
Then:
:$\sigma_h \le \max \set {\sigma_f, \sigma_g}$ | Follows from Dirichlet Series of Convolution of Arithmetic Functions
{{ProofWanted}}
Category:Dirichlet Series
e03r49gdkg8ry56gqsgk754xuk3dgql | Let $f, g: \N \to \C$ be [[Definition:Arithmetic Function|arithmetic functions]] with [[Definition:Dirichlet Convolution|Dirichlet convolution]] $h = f * g$.
Let $F, G, H$ be their [[Definition:Dirichlet Series|Dirichlet series]].
Let $\sigma_f, \sigma_g, \sigma_h$ be their [[Definition:Abscissa of Absolute Convergen... | Follows from [[Dirichlet Series of Convolution of Arithmetic Functions]]
{{ProofWanted}}
[[Category:Dirichlet Series]]
e03r49gdkg8ry56gqsgk754xuk3dgql | Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series | https://proofwiki.org/wiki/Upper_Bound_for_Abscissa_of_Absolute_Convergence_of_Product_of_Dirichlet_Series | https://proofwiki.org/wiki/Upper_Bound_for_Abscissa_of_Absolute_Convergence_of_Product_of_Dirichlet_Series | [
"Dirichlet Series"
] | [
"Definition:Arithmetic Function",
"Definition:Dirichlet Convolution",
"Definition:Dirichlet Series",
"Definition:Abscissa of Absolute Convergence"
] | [
"Dirichlet Series of Convolution of Arithmetic Functions",
"Category:Dirichlet Series"
] |
proofwiki-13745 | Riemann Zeta Function in terms of Dirichlet Eta Function | Let $\zeta$ be the Riemann zeta function.
Let $\eta$ be the Dirichlet eta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then:
:$\map \zeta s = \dfrac 1 {1 - 2^{1 - s} } \map \eta s$ | {{begin-eqn}}
{{eqn | l = \map \zeta s - \map \eta s
| r = \sum_{n \mathop = 1}^\infty \frac 1 {n^s} - \sum_{n \mathop = 1}^\infty \frac{\paren {-1}^{n - 1} } {n^s}
| c = {{Defof|Riemann Zeta Function}}, {{Defof|Dirichlet Eta Function}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \paren {\frac 1 {n^s} + \fra... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\eta$ be the [[Definition:Dirichlet Eta Function|Dirichlet eta function]].
Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma > 1$.
Then:
:$\map \zeta s = \dfrac 1 {1 - 2... | {{begin-eqn}}
{{eqn | l = \map \zeta s - \map \eta s
| r = \sum_{n \mathop = 1}^\infty \frac 1 {n^s} - \sum_{n \mathop = 1}^\infty \frac{\paren {-1}^{n - 1} } {n^s}
| c = {{Defof|Riemann Zeta Function}}, {{Defof|Dirichlet Eta Function}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \paren {\frac 1 {n^s} + \fra... | Riemann Zeta Function in terms of Dirichlet Eta Function | https://proofwiki.org/wiki/Riemann_Zeta_Function_in_terms_of_Dirichlet_Eta_Function | https://proofwiki.org/wiki/Riemann_Zeta_Function_in_terms_of_Dirichlet_Eta_Function | [
"Riemann Zeta Function",
"Dirichlet Eta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Dirichlet Eta Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part"
] | [
"Sum of Summations equals Summation of Sum"
] |
proofwiki-13746 | Functional Equation for Riemann Zeta Function | Let $\zeta$ be the Riemann zeta function.
Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$.
Then:
:$\pi^{-s/2 } \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$
where $\Gamma$ is the gamma function | Let $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$.
Then from Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function we have:
:$(1): \quad \ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \int_1^\infty \paren {x^{s / 2 - 1} + x^{- s / 2... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\map \zeta s$ have an [[Definition:Analytic Continuation|analytic continuation]] for $\map \Re s > 0$.
Then:
:$\pi^{-s/2 } \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {... | Let $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$.
Then from [[Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function]] we have:
:$(1): \quad \ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \int_1^\infty \paren {x^{s / 2 - 1} + x^{-... | Functional Equation for Riemann Zeta Function | https://proofwiki.org/wiki/Functional_Equation_for_Riemann_Zeta_Function | https://proofwiki.org/wiki/Functional_Equation_for_Riemann_Zeta_Function | [
"Riemann Zeta Function",
"Reflection Formulas"
] | [
"Definition:Riemann Zeta Function",
"Definition:Analytic Continuation",
"Definition:Gamma Function"
] | [
"Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function"
] |
proofwiki-13747 | Integral Representation of Riemann Zeta Function in terms of Fractional Part | Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then
:$\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
where $\fractpart x$ denotes the fractional part of $x$. | First, we observe that:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {n - 1} {n^s}
| r = \sum_{n \mathop = 0}^\infty \dfrac n {\paren {n + 1}^s}
| c = {{Corollary|Translation of Index Variable of Summation}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac n {\paren {n + 1}^s}
| c = $... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\sigma > 1$.
Then
:$\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
where $\fractpart x$ denot... | First, we observe that:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {n - 1} {n^s}
| r = \sum_{n \mathop = 0}^\infty \dfrac n {\paren {n + 1}^s}
| c = {{Corollary|Translation of Index Variable of Summation}}
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac n {\paren {n + 1}^s}
| c = $... | Integral Representation of Riemann Zeta Function in terms of Fractional Part | https://proofwiki.org/wiki/Integral_Representation_of_Riemann_Zeta_Function_in_terms_of_Fractional_Part | https://proofwiki.org/wiki/Integral_Representation_of_Riemann_Zeta_Function_in_terms_of_Fractional_Part | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Fractional Part"
] | [
"Definition:Vanish",
"Definition:Floor Function",
"Linear Combination of Complex Integrals",
"Primitive of Power"
] |
proofwiki-13748 | Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function | Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\map \Re s > 1$.
Then:
$\ds \map \zeta s = \frac 1 {1 - 2^{1 - s} } \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2^{n + 1} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom n k \paren {k + 1}^{-s} }$ | Use Riemann Zeta Function in terms of Dirichlet Eta Function and Binomial Theorem.
{{ProofWanted}}
Category:Riemann Zeta Function
hms61q95kwv0fzzoq7ei3khzdgyap9t | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $s \in \C$ be a [[Definition:Complex Number|complex number]] with [[Definition:Real Part|real part]] $\map \Re s > 1$.
Then:
$\ds \map \zeta s = \frac 1 {1 - 2^{1 - s} } \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2^{n + 1} } \sum_{k... | Use [[Riemann Zeta Function in terms of Dirichlet Eta Function]] and [[Binomial Theorem]].
{{ProofWanted}}
[[Category:Riemann Zeta Function]]
hms61q95kwv0fzzoq7ei3khzdgyap9t | Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function | https://proofwiki.org/wiki/Binomial_Form_of_Relation_between_Riemann_Zeta_Function_and_Dirichlet_Eta_Function | https://proofwiki.org/wiki/Binomial_Form_of_Relation_between_Riemann_Zeta_Function_and_Dirichlet_Eta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Complex Number",
"Definition:Complex Number/Real Part"
] | [
"Riemann Zeta Function in terms of Dirichlet Eta Function",
"Binomial Theorem",
"Category:Riemann Zeta Function"
] |
proofwiki-13749 | Analytic Continuations of Riemann Zeta Function to Complex Plane | The Riemann zeta function $\zeta$ has a unique analytic continuation to $\C \setminus \set 1$. | Note that by Riemann Zeta Function is Analytic, $\map \zeta s$ is indeed analytic for $\map \Re s > 1$.
By Complex Plane minus Point is Connected, $\C \setminus \set 1$ is connected.
By Uniqueness of Analytic Continuation, there is at most one analytic continuation of $\zeta$ to $\C \setminus \set 1$.
{{finish|link to ... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] $\zeta$ has a [[Definition:Unique|unique]] [[Definition:Analytic Continuation|analytic continuation]] to $\C \setminus \set 1$. | Note that by [[Riemann Zeta Function is Analytic]], $\map \zeta s$ is indeed [[Definition:Analytic Complex Function|analytic]] for $\map \Re s > 1$.
By [[Complex Plane minus Point is Connected]], $\C \setminus \set 1$ is [[Definition:Connected Subset of Complex Plane|connected]].
By [[Uniqueness of Analytic Continuat... | Analytic Continuations of Riemann Zeta Function to Complex Plane | https://proofwiki.org/wiki/Analytic_Continuations_of_Riemann_Zeta_Function_to_Complex_Plane | https://proofwiki.org/wiki/Analytic_Continuations_of_Riemann_Zeta_Function_to_Complex_Plane | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Unique",
"Definition:Analytic Continuation"
] | [
"Riemann Zeta Function is Analytic",
"Definition:Analytic Function/Complex Plane",
"Complex Plane minus Point is Connected",
"Definition:Connected Subset of Complex Plane",
"Uniqueness of Analytic Continuation",
"Definition:Analytic Continuation",
"Functional Equation for Riemann Zeta Function",
"Anal... |
proofwiki-13750 | Analytic Continuations of Riemann Zeta Function to Right Half-Plane | The Riemann zeta function has a unique analytic continuation to $\set {s \in \C : \map \Re s > 0} \setminus \set 1$, the half-plane $\map \Re s > 0$ minus the point $s = 1$. | Note that by Riemann Zeta Function is Analytic, $\map \zeta s$ is indeed analytic for $\map \Re s > 1$.
By Complex Half-Plane minus Point is Connected, $\set {\sigma > 0} \setminus \set 1$ is connected.
By Uniqueness of Analytic Continuation, there is at most one analytic continuation of $\zeta$ to $\set {\sigma > 0} \... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] has a [[Definition:Unique|unique]] [[Definition:Analytic Continuation|analytic continuation]] to $\set {s \in \C : \map \Re s > 0} \setminus \set 1$, the [[Definition:Complex Half-Plane|half-plane]] $\map \Re s > 0$ minus the point $s = 1$. | Note that by [[Riemann Zeta Function is Analytic]], $\map \zeta s$ is indeed [[Definition:Analytic Complex Function|analytic]] for $\map \Re s > 1$.
By [[Complex Half-Plane minus Point is Connected]], $\set {\sigma > 0} \setminus \set 1$ is [[Definition:Connected Subset of Complex Plane|connected]].
By [[Uniqueness o... | Analytic Continuations of Riemann Zeta Function to Right Half-Plane | https://proofwiki.org/wiki/Analytic_Continuations_of_Riemann_Zeta_Function_to_Right_Half-Plane | https://proofwiki.org/wiki/Analytic_Continuations_of_Riemann_Zeta_Function_to_Right_Half-Plane | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Unique",
"Definition:Analytic Continuation",
"Definition:Complex Half-Plane"
] | [
"Riemann Zeta Function is Analytic",
"Definition:Analytic Function/Complex Plane",
"Complex Half-Plane minus Point is Connected",
"Definition:Connected Subset of Complex Plane",
"Uniqueness of Analytic Continuation",
"Definition:Analytic Continuation",
"Analytic Continuation of Riemann Zeta Function usi... |
proofwiki-13751 | Functional Equation for Completed Riemann Zeta Function | Let $\xi : \C \to \C$ be the completed Riemann zeta function.
Let $s\in \C$ be a complex number.
Then:
:$\map \xi s = \map \xi {1 - s}$ | {{ProofWanted|Use Functional Equation for Riemann Zeta Function.}}
Category:Riemann Zeta Function
Category:Reflection Formulas
b6817fdc3ocmub9axym4500feep3v52 | Let $\xi : \C \to \C$ be the [[Definition:Completed Riemann Zeta Function|completed Riemann zeta function]].
Let $s\in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\map \xi s = \map \xi {1 - s}$ | {{ProofWanted|Use [[Functional Equation for Riemann Zeta Function]].}}
[[Category:Riemann Zeta Function]]
[[Category:Reflection Formulas]]
b6817fdc3ocmub9axym4500feep3v52 | Functional Equation for Completed Riemann Zeta Function | https://proofwiki.org/wiki/Functional_Equation_for_Completed_Riemann_Zeta_Function | https://proofwiki.org/wiki/Functional_Equation_for_Completed_Riemann_Zeta_Function | [
"Riemann Zeta Function",
"Reflection Formulas"
] | [
"Definition:Completed Riemann Zeta Function",
"Definition:Complex Number"
] | [
"Functional Equation for Riemann Zeta Function",
"Category:Riemann Zeta Function",
"Category:Reflection Formulas"
] |
proofwiki-13752 | Dirichlet Series of Inverse of Arithmetic Function | Let $f : \N \to\C$ be an arithmetic function.
Let $g : \N \to \C$ be an Dirichlet inverse of $f$.
Let $F, G$ be their Dirichlet series.
Let $s \in \C$ such that both $\map F s$ and $\map G s$ converge absolutely.
Then $\map F s \cdot \map G s = 1$. | Let $\varepsilon$ be the identity arithmetic function.
By Dirichlet Series of Identity Arithmetic Function, $\varepsilon$ has Dirichlet series $\map E s = 1$.
By Dirichlet Series of Convolution of Arithmetic Functions, $\map F s \map G s = 1$.
{{qed}}
Category:Dirichlet Series
339fjb352cdslw7atl3gk5041grpmt7 | Let $f : \N \to\C$ be an [[Definition:Arithmetic Function|arithmetic function]].
Let $g : \N \to \C$ be an [[Definition:Dirichlet Inverse of Arithmetic Function|Dirichlet inverse]] of $f$.
Let $F, G$ be their [[Definition:Dirichlet Series|Dirichlet series]].
Let $s \in \C$ such that both $\map F s$ and $\map G s$ [[... | Let $\varepsilon$ be the [[Definition:Identity Arithmetic Function|identity arithmetic function]].
By [[Dirichlet Series of Identity Arithmetic Function]], $\varepsilon$ has [[Definition:Dirichlet Series|Dirichlet series]] $\map E s = 1$.
By [[Dirichlet Series of Convolution of Arithmetic Functions]], $\map F s \map ... | Dirichlet Series of Inverse of Arithmetic Function | https://proofwiki.org/wiki/Dirichlet_Series_of_Inverse_of_Arithmetic_Function | https://proofwiki.org/wiki/Dirichlet_Series_of_Inverse_of_Arithmetic_Function | [
"Dirichlet Series"
] | [
"Definition:Arithmetic Function",
"Definition:Dirichlet Inverse of Arithmetic Function",
"Definition:Dirichlet Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Identity Arithmetic Function",
"Dirichlet Series of Identity Arithmetic Function",
"Definition:Dirichlet Series",
"Dirichlet Series of Convolution of Arithmetic Functions",
"Category:Dirichlet Series"
] |
proofwiki-13753 | Invertibility of Arithmetic Functions | Let $f: \N \to \C$ be an arithmetic function.
Then $f$ has a Dirichlet inverse {{iff}}:
:$\map f 1 \ne 0$ | Let $*$ denote Dirichlet convolution.
Let $\varepsilon$ denote the identity arithmetic function. | Let $f: \N \to \C$ be an [[Definition:Arithmetic Function|arithmetic function]].
Then $f$ has a [[Definition:Dirichlet Inverse of Arithmetic Function|Dirichlet inverse]] {{iff}}:
:$\map f 1 \ne 0$ | Let $*$ denote [[Definition:Dirichlet Convolution|Dirichlet convolution]].
Let $\varepsilon$ denote the [[Definition:Identity Arithmetic Function|identity arithmetic function]]. | Invertibility of Arithmetic Functions | https://proofwiki.org/wiki/Invertibility_of_Arithmetic_Functions | https://proofwiki.org/wiki/Invertibility_of_Arithmetic_Functions | [
"Arithmetic Functions"
] | [
"Definition:Arithmetic Function",
"Definition:Dirichlet Inverse of Arithmetic Function"
] | [
"Definition:Dirichlet Convolution",
"Definition:Identity Arithmetic Function"
] |
proofwiki-13754 | Equivalence of Definitions of Consistent Set of Formulas | {{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
Let $\LL_0$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$.
Let $\FF$ be a collection of logical formulas. | === Definition 1 implies Definition 2 ===
Suppose that $\FF \not \vdash_{\mathscr P} \phi$.
{{AimForCont}} that there exists $\psi$ such that $\FF \vdash_{\mathscr P} \psi$ and $\FF \vdash_{\mathscr P} \neg \psi$.
Then by the Rule of Explosion (Variant 3):
:$\psi, \neg \psi \vdash_{\mathscr P} \phi$
and therefore, comb... | {{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL_0$.
Let $\FF$ be a coll... | === Definition 1 implies Definition 2 ===
Suppose that $\FF \not \vdash_{\mathscr P} \phi$.
{{AimForCont}} that there exists $\psi$ such that $\FF \vdash_{\mathscr P} \psi$ and $\FF \vdash_{\mathscr P} \neg \psi$.
Then by the [[Rule of Explosion/Variant 3|Rule of Explosion (Variant 3)]]:
:$\psi, \neg \psi \vdash_{\... | Equivalence of Definitions of Consistent Set of Formulas | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Consistent_Set_of_Formulas | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Consistent_Set_of_Formulas | [
"Proof Systems",
"Language of Propositional Logic"
] | [
"Definition:Language of Propositional Logic",
"Definition:Proof System",
"Definition:Logical Formula"
] | [
"Rule of Explosion/Variant 3",
"Definition:Contradiction"
] |
proofwiki-13755 | Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part | Let $\zeta$ denote the Riemann zeta function.
The analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ is given by:
:$\ds \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
where:
:$x^{-s - 1}$ takes the principal value $e^{-\map \ln x \paren {s + 1} }$
:$\fractpart x$ is the fractional par... | Let $s = \sigma + i t$.
By Integral Representation of Riemann Zeta Function in terms of Fractional Part, the above integral coincides with $\map \zeta s$ for $\sigma > 1$.
We show that it is analytic for $0 < \sigma < 1$.
For $n \ge 1$, let:
{{begin-eqn}}
{{eqn | l = \cmod {a_n}
| r = \cmod {\int_n^{n + 1} s \fra... | Let $\zeta$ denote the [[Definition:Riemann Zeta Function|Riemann zeta function]].
The [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to the [[Definition:Complex Halfplane|half-plane]] $\map \Re s > 0$ is given by:
:$\ds \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
where:
:$... | Let $s = \sigma + i t$.
By [[Integral Representation of Riemann Zeta Function in terms of Fractional Part]], the above integral coincides with $\map \zeta s$ for $\sigma > 1$.
We show that it is [[Definition:Analytic Complex Function|analytic]] for $0 < \sigma < 1$.
For $n \ge 1$, let:
{{begin-eqn}}
{{eqn | l = \c... | Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part | https://proofwiki.org/wiki/Analytic_Continuation_of_Riemann_Zeta_Function_using_Mellin_Transform_of_Fractional_Part | https://proofwiki.org/wiki/Analytic_Continuation_of_Riemann_Zeta_Function_using_Mellin_Transform_of_Fractional_Part | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Analytic Continuation",
"Definition:Complex Halfplane",
"Definition:Multifunction/Principal Branch",
"Definition:Fractional Part"
] | [
"Integral Representation of Riemann Zeta Function in terms of Fractional Part",
"Definition:Analytic Function/Complex Plane",
"Triangle Inequality for Integrals/Complex",
"Mean Value Theorem",
"Definition:Compact Topological Space/Subspace",
"Definition:Subset",
"Definition:Complex Halfplane",
"Defini... |
proofwiki-13756 | Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function | Let $\zeta$ be the Riemann zeta function.
Let $\eta$ be the Dirichlet Eta Function.
Then:
:$\dfrac 1 {1 - 2^{1 - s} } \map \eta s$
defines an analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ minus $s = 1$. | By Riemann Zeta Function in terms of Dirichlet Eta Function, it coincides with $\zeta$ for $\map \Re s > 1$.
By Dirichlet Eta Function is Analytic, it is analytic for $\map \Re s > 0$, except at $s = 1$.
{{qed}} | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\eta$ be the [[Definition:Dirichlet Eta Function|Dirichlet Eta Function]].
Then:
:$\dfrac 1 {1 - 2^{1 - s} } \map \eta s$
defines an [[Definition:Analytic Continuation|analytic continuation]] of $\zeta$ to the [[Definition:Complex Ha... | By [[Riemann Zeta Function in terms of Dirichlet Eta Function]], it coincides with $\zeta$ for $\map \Re s > 1$.
By [[Dirichlet Eta Function is Analytic]], it is [[Definition:Analytic Complex Function|analytic]] for $\map \Re s > 0$, except at $s = 1$.
{{qed}} | Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function | https://proofwiki.org/wiki/Analytic_Continuation_of_Riemann_Zeta_Function_using_Dirichlet_Eta_Function | https://proofwiki.org/wiki/Analytic_Continuation_of_Riemann_Zeta_Function_using_Dirichlet_Eta_Function | [
"Riemann Zeta Function",
"Dirichlet Eta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Dirichlet Eta Function",
"Definition:Analytic Continuation",
"Definition:Complex Halfplane"
] | [
"Riemann Zeta Function in terms of Dirichlet Eta Function",
"Dirichlet Eta Function is Analytic",
"Definition:Analytic Function/Complex Plane"
] |
proofwiki-13757 | Complex Modulus of Real Number equals Absolute Value | Let $x \in \R$ be a real number.
Then the complex modulus of $x$ equals the absolute value of $x$. | Let $x = x + 0 i \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {x + 0 i}
| r = \sqrt {x^2 + 0^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sqrt {x^2}
| c =
}}
{{eqn | r = \size x
| c = {{Defof|Absolute Value|index = 2}}
}}
{{end-eqn}}
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then the [[Definition:Complex Modulus|complex modulus]] of $x$ equals the [[Definition:Absolute Value of Real Number|absolute value]] of $x$. | Let $x = x + 0 i \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {x + 0 i}
| r = \sqrt {x^2 + 0^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sqrt {x^2}
| c =
}}
{{eqn | r = \size x
| c = {{Defof|Absolute Value|index = 2}}
}}
{{end-eqn}}
{{qed}} | Complex Modulus of Real Number equals Absolute Value | https://proofwiki.org/wiki/Complex_Modulus_of_Real_Number_equals_Absolute_Value | https://proofwiki.org/wiki/Complex_Modulus_of_Real_Number_equals_Absolute_Value | [
"Complex Modulus",
"Absolute Value Function"
] | [
"Definition:Real Number",
"Definition:Complex Modulus",
"Definition:Absolute Value"
] | [] |
proofwiki-13758 | Equivalence of Definitions of Ceiling Function | Let $x$ be a real number.
{{TFAE|def = Ceiling Function}} | === Definition 1 equals Definition 2 ===
Follows from Infimum of Set of Integers equals Smallest Element.
{{qed|lemma}} | Let $x$ be a [[Definition:Real Number|real number]].
{{TFAE|def = Ceiling Function}} | === Definition 1 equals Definition 2 ===
Follows from [[Infimum of Set of Integers equals Smallest Element]].
{{qed|lemma}} | Equivalence of Definitions of Ceiling Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ceiling_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ceiling_Function | [
"Ceiling Function"
] | [
"Definition:Real Number"
] | [
"Infimum of Set of Integers equals Smallest Element",
"Infimum of Set of Integers equals Smallest Element"
] |
proofwiki-13759 | Characterizing Property of Infimum of Subset of Real Numbers | Let $S \subset \R$ be a non-empty subset of the real numbers.
Let $S$ be bounded below.
Let $\alpha \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\alpha$ is the infimum of $S$}}
{{item|(2):|$\alpha$ is a lower bound for $S$ and $\forall \epsilon \in \R_{> 0}: \exists x \in S: x < \alpha + \epsilon$}}
{{end-itemize}} | === $(1)$ implies $(2)$ ===
Let $\alpha$ be the infimum of $S$.
Then by definition, $\alpha$ is a lower bound for $S$.
Let $\epsilon>0$.
Because $\alpha+\epsilon>\alpha$, it is not a lower bound for $S$.
Thus there exists $x\in S$ with $x < \alpha + \epsilon$.
{{qed|lemma}} | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Let $S$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Let $\alpha \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\alpha$ is the [[Definition:Infimum ... | === $(1)$ implies $(2)$ ===
Let $\alpha$ be the [[Definition:Infimum of Subset of Real Numbers|infimum]] of $S$.
Then by definition, $\alpha$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for $S$.
Let $\epsilon>0$.
Because $\alpha+\epsilon>\alpha$, it is not a [[Definition:Lower Bound of Sub... | Characterizing Property of Infimum of Subset of Real Numbers | https://proofwiki.org/wiki/Characterizing_Property_of_Infimum_of_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Characterizing_Property_of_Infimum_of_Subset_of_Real_Numbers | [
"Infima",
"Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers"
] | [
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers"
] |
proofwiki-13760 | Characterizing Property of Supremum of Subset of Real Numbers | Let $S \subset \R$ be a non-empty subset of the real numbers.
Let $S$ be bounded above.
Let $\omega \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\omega$ is the supremum of $S$}}
{{item|(2):|$\omega$ is an upper bound for $S$ and $\forall \epsilon \in \R_{> 0}: \exists x \in S: x > \omega - \epsilon$}}
{{end-itemize... | === $(1)$ implies $(2)$ ===
Let $\omega$ be the supremum of $S$.
Then by definition, $\omega$ is an upper bound for $S$.
Let $\epsilon > 0$.
Because $\omega - \epsilon < \omega$, it is not an upper bound for $S$.
Thus there exists $x\in S$ with $x > \omega - \epsilon$.
{{qed|lemma}} | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Let $\omega \in \R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$\omega$ is the [[Definition:Supremum... | === $(1)$ implies $(2)$ ===
Let $\omega$ be the [[Definition:Supremum of Subset of Real Numbers|supremum]] of $S$.
Then by definition, $\omega$ is an [[Definition:upper Bound of Subset of Real Numbers|upper bound]] for $S$.
Let $\epsilon > 0$.
Because $\omega - \epsilon < \omega$, it is not an [[Definition:Upper Bo... | Characterizing Property of Supremum of Subset of Real Numbers | https://proofwiki.org/wiki/Characterizing_Property_of_Supremum_of_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Characterizing_Property_of_Supremum_of_Subset_of_Real_Numbers | [
"Suprema",
"Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers"
] | [
"Definition:Supremum of Set/Real Numbers",
"Definition:upper Bound of Subset of Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:upper Bound of Subset of Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers"
] |
proofwiki-13761 | Limit of Positive Real Sequence is Positive | Let $\sequence {x_n}$ be a sequence of positive real numbers.
Let $x_n$ converge to $L$.
Then $L \ge 0$. | {{AimForCont}} $L < 0$.
Then for any $n \in \N$:
{{begin-eqn}}
{{eqn | l = \size {x_n - L}
| r = x_n - L
| c = $x_n \ge 0 > L$
}}
{{eqn | o = \ge
| r = -L
| c = $> 0$
}}
{{end-eqn}}
This contradicts {{Defof|Convergent Real Sequence}}.
Hence we must have $L \ge 0$.
{{qed}}
Category:Convergence
px... | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Positive Real Number|positive real numbers]].
Let $x_n$ [[Definition:Convergent Real Sequence|converge]] to $L$.
Then $L \ge 0$. | {{AimForCont}} $L < 0$.
Then for any $n \in \N$:
{{begin-eqn}}
{{eqn | l = \size {x_n - L}
| r = x_n - L
| c = $x_n \ge 0 > L$
}}
{{eqn | o = \ge
| r = -L
| c = $> 0$
}}
{{end-eqn}}
This contradicts {{Defof|Convergent Real Sequence}}.
Hence we must have $L \ge 0$.
{{qed}}
[[Category:Conver... | Limit of Positive Real Sequence is Positive | https://proofwiki.org/wiki/Limit_of_Positive_Real_Sequence_is_Positive | https://proofwiki.org/wiki/Limit_of_Positive_Real_Sequence_is_Positive | [
"Convergence"
] | [
"Definition:Sequence",
"Definition:Positive/Real Number",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Category:Convergence"
] |
proofwiki-13762 | Real Sequence with Nonzero Limit is Eventually Nonzero | Let $\sequence {x_n}$ be a real sequence.
Let $\sequence {x_n}$ converge to $a \ne 0$.
Then:
:$\exists N \in \N: \forall n \ge N: x_n \ne 0$
That is, eventually every term of $\sequence {x_n}$ becomes non-zero. | Suppose $a > 0$.
By Sequence Converges to Within Half Limit:
:$\exists N \in \N: \forall n > N: x_n > \dfrac a 2 > 0$
Now suppose $a < 0$.
By Sequence Converges to Within Half Limit:
:$\exists N \in \N: \forall n > N: x_n < \dfrac a 2 < 0$
This shows that if $a \ne 0$:
:$\exists N \in \N: \forall n > N: x_n \ne 0$
{{... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|real sequence]].
Let $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converge]] to $a \ne 0$.
Then:
:$\exists N \in \N: \forall n \ge N: x_n \ne 0$
That is, eventually every term of $\sequence {x_n}$ becomes non-zero. | Suppose $a > 0$.
By [[Sequence Converges to Within Half Limit]]:
:$\exists N \in \N: \forall n > N: x_n > \dfrac a 2 > 0$
Now suppose $a < 0$.
By [[Sequence Converges to Within Half Limit]]:
:$\exists N \in \N: \forall n > N: x_n < \dfrac a 2 < 0$
This shows that if $a \ne 0$:
:$\exists N \in \N: \forall n >... | Real Sequence with Nonzero Limit is Eventually Nonzero | https://proofwiki.org/wiki/Real_Sequence_with_Nonzero_Limit_is_Eventually_Nonzero | https://proofwiki.org/wiki/Real_Sequence_with_Nonzero_Limit_is_Eventually_Nonzero | [
"Convergence"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Sequence Converges to Within Half Limit",
"Sequence Converges to Within Half Limit"
] |
proofwiki-13763 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|1}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \cot {\frac {\pi m} {2 n} }
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {2 k n + m} - \frac 1 {2 k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {2 n +... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|1}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \cot {\frac {\pi m} {2 n} }
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {2 k n + m} - \frac 1 {2 k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {2 n ... | Leibniz's Formula for Pi/Proof by Mittag-Leffler Expansion for Cotangent Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Mittag-Leffler_Expansion_for_Cotangent_Function | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [] |
proofwiki-13764 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | Let $\LL$ denote the logarithmic derivative.
On the open set $\C \setminus \Z$ we have:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \map \LL {\map \sin {\pi z} }
| c = Primitive of Cotangent Function, or a complex version thereof
}}
{{eqn | r = \map \LL {\pi z \prod_{n \mathop = 1}^\infty \paren {1 - \f... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | Let $\LL$ denote the [[Definition:Logarithmic Derivative of Meromorphic Function|logarithmic derivative]].
On the [[Definition:Open Subset of Complex Plane|open set]] $\C \setminus \Z$ we have:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \map \LL {\map \sin {\pi z} }
| c = [[Primitive of Cotangent Fun... | Mittag-Leffler Expansion for Cotangent Function/Proof 1 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_1 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Definition:Logarithmic Derivative of Meromorphic Function",
"Definition:Open Set/Complex Analysis",
"Primitive of Cotangent Function",
"Euler Formula for Sine Function",
"Logarithmic Derivative of Infinite Product of Analytic Functions",
"Power Rule for Derivatives"
] |
proofwiki-13765 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | Let $\map \zeta s$ be the Riemann zeta function.
Let $\ds \map g z = \sum_{n \mathop = 1}^\infty z^n \map \zeta {2 n}$ be the generating function of $\map \zeta {2 n}$
By Power Series Expansion for Cotangent Function, for $\size z < 1$:
{{begin-eqn}}
{{eqn | l = \pi \map \cot {\pi z}
| r = \sum_{n \mathop = 0}^\... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | Let $\map \zeta s$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\ds \map g z = \sum_{n \mathop = 1}^\infty z^n \map \zeta {2 n}$ be the [[Definition:Generating Function| generating function]] of $\map \zeta {2 n}$
By [[Power Series Expansion for Cotangent Function]], for $\size z < 1$:
{{b... | Mittag-Leffler Expansion for Cotangent Function/Proof 2 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_2 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Generating Function",
"Power Series Expansion for Cotangent Function",
"Riemann Zeta Function at Even Integers",
"Analytic Continuation of Generating Function of Dirichlet Series",
"Uniqueness of Analytic Continuation"
] |
proofwiki-13766 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | Define the following complex functions.
{{begin-eqn}}
{{eqn | l = \map g z
| r = \lim_{N \mathop \to \infty} \sum_{n \mathop = -N}^N \frac 1 {z - n}
| c =
}}
{{eqn | r = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}
| c =
}}
{{eqn | l = \map h z
| r = \lim_{s \mathop \to z} \p... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | Define the following [[Definition:Complex Function|complex functions]].
{{begin-eqn}}
{{eqn | l = \map g z
| r = \lim_{N \mathop \to \infty} \sum_{n \mathop = -N}^N \frac 1 {z - n}
| c =
}}
{{eqn | r = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}
| c =
}}
{{eqn | l = \map h z
... | Mittag-Leffler Expansion for Cotangent Function/Proof 3 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_3 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Definition:Complex Function",
"Comparison Test",
"Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1",
"Uniform Limit of Analytic Functions is Analytic",
"Definition:Analytic Function/Complex Plane",
"Definition:Open Set/Complex Analysis",
"Definition:Cotangent/Complex Functi... |
proofwiki-13767 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | We have that:
:$\cot \pi x = \dfrac {\cos \pi x} {\sin \pi x}$
has a denominator which is $0$ at $x = 0, \pm 1, \pm 2, \ldots$.
Hence the limitation on the domain of $x \cot \pi x$ to exclude integer $x$.
Having established that, we should be able to express $\cot \pi x$ in the form:
:$\cot \pi x = \dfrac a x + \ds \su... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | We have that:
:$\cot \pi x = \dfrac {\cos \pi x} {\sin \pi x}$
has a [[Definition:Denominator|denominator]] which is $0$ at $x = 0, \pm 1, \pm 2, \ldots$.
Hence the limitation on the [[Definition:Domain of Mapping|domain]] of $x \cot \pi x$ to exclude [[Definition:Integer|integer]] $x$.
Having established that, we sh... | Mittag-Leffler Expansion for Cotangent Function/Proof 4 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_4 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Definition:Fraction/Denominator",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Integer",
"Definition:Partial Fractions Expansion"
] |
proofwiki-13768 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | From the Euler Formula for Sine Function:
:$\ds \sin x = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }$
Taking the logarithm of both sides:
{{begin-eqn}}
{{eqn | l = \map \ln {\sin x}
| r = \ln x + \sum_{n \mathop = 1}^\infty \map \ln {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | From the [[Euler Formula for Sine Function]]:
:$\ds \sin x = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }$
Taking the [[Definition:Logarithm|logarithm]] of both sides:
{{begin-eqn}}
{{eqn | l = \map \ln {\sin x}
| r = \ln x + \sum_{n \mathop = 1}^\infty \map \ln {1 - \frac {x^2} {n^2 \pi... | Mittag-Leffler Expansion for Cotangent Function/Proof 5 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_5 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Euler Formula for Sine Function",
"Definition:Logarithm",
"Definition:Differentiation",
"Derivative of Composite Function",
"Definition:Fraction/Denominator"
] |
proofwiki-13769 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | From Euler's Reflection Formula:
:$\forall x \notin \Z: \map \Gamma x \map \Gamma {1 - x} = \dfrac \pi {\map \sin {\pi x} }$
Taking the logarithm of both sides:
{{begin-eqn}}
{{eqn | l = \map \ln {\map {\Gamma} x } + \map \ln {\map {\Gamma} {1 - x} }
| r = \map \ln {\pi } - \map \ln {\map \sin {\pi x} }
| c... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | From [[Euler's Reflection Formula]]:
:$\forall x \notin \Z: \map \Gamma x \map \Gamma {1 - x} = \dfrac \pi {\map \sin {\pi x} }$
Taking the [[Definition:Natural Logarithm|logarithm]] of both sides:
{{begin-eqn}}
{{eqn | l = \map \ln {\map {\Gamma} x } + \map \ln {\map {\Gamma} {1 - x} }
| r = \map \ln {\pi } - \... | Mittag-Leffler Expansion for Cotangent Function/Proof 6 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_6 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Euler's Reflection Formula",
"Definition:Natural Logarithm",
"Sum of Logarithms/Natural Logarithm",
"Difference of Logarithms",
"Definition:Derivative",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Derivative of Sine Function",
"Reciprocal times Derivative of Gam... |
proofwiki-13770 | Mittag-Leffler Expansion for Cotangent Function | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an integer
:$\cot$ is the cotangent function. | Following the method of Herglotz, define $/map f x = \pi \cot \pi x$.
This $f$ is seen to obey four important properties:
:$1$) It is continuous on its domain $\R\setminus \Z$,
:$2$) It is periodic with period $1$,
:$3$) It obeys the functional equation $f(\frac x 2) + f(\frac{x+1} 2) = 2f(x)$, and
:$4$) It holds ... | :$\ds \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]]
:$\cot$ is the [[Definition:Complex Cotangent Function|cotangent function]]. | Following the method of Herglotz, define $/map f x = \pi \cot \pi x$.
This $f$ is seen to obey four important properties:
:$1$) It is [[Definition:Continuous Real Function at Point|continuous]] on its domain $\R\setminus \Z$,
:$2$) It is [[Definition:Periodic|periodic]] with [[Definition:Periodic Real Function/Pe... | Mittag-Leffler Expansion for Cotangent Function/Proof 7 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cotangent_Function/Proof_7 | [
"Mittag-Leffler Expansion for Cotangent Function",
"Cotangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Integer",
"Definition:Cotangent/Complex Function"
] | [
"Definition:Continuous Real Function/Point",
"Definition:Periodic",
"Definition:Periodic Real Function/Period",
"Definition:Functional Equation",
"Definition:Partial Fractions Expansion",
"Comparison Test",
"Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1",
"Definition:Co... |
proofwiki-13771 | Laurent Series Expansion for Cotangent Function | {{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}
}}
{{eqn | r = \frac 1 z - 2 \paren {\dfrac {\pi^2 } 6 z + \dfrac {\pi^4 } {90 } z^3 + \dfrac {\pi^6 } {945 } z^5 + \cdots}
}}
{{eqn | r = \frac 1 z - \dfrac {\pi^2 } 3 z - \dfrac {\pi^4 } {45 } z... | From Mittag-Leffler Expansion for Cotangent Function:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \frac z {z^2 - k^2}
}}
{{end-eqn}}
Factoring $-\dfrac 1 {k^2}$:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \paren {\fr... | {{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}
}}
{{eqn | r = \frac 1 z - 2 \paren {\dfrac {\pi^2 } 6 z + \dfrac {\pi^4 } {90 } z^3 + \dfrac {\pi^6 } {945 } z^5 + \cdots}
}}
{{eqn | r = \frac 1 z - \dfrac {\pi^2 } 3 z - \dfrac {\pi^4 } {45 } z... | From [[Mittag-Leffler Expansion for Cotangent Function]]:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \frac z {z^2 - k^2}
}}
{{end-eqn}}
Factoring $-\dfrac 1 {k^2}$:
{{begin-eqn}}
{{eqn | l = \pi \cot \pi z
| r = \frac 1 z + 2 \sum_{k \mathop = 1}^\infty \pa... | Laurent Series Expansion for Cotangent Function | https://proofwiki.org/wiki/Laurent_Series_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Laurent_Series_Expansion_for_Cotangent_Function | [
"Laurent Series Expansion for Cotangent Function",
"Laurent Series Expansions",
"Cotangent Function",
"Riemann Zeta Function at Even Integers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Mittag-Leffler Expansion for Cotangent Function",
"Sum of Infinite Geometric Sequence",
"Product of Absolutely Convergent Series"
] |
proofwiki-13772 | Equivalence of Definitions of Removable Discontinuity of Real Function | Let $A \subseteq \R$ be a subset of the real numbers.
Let $f: A \to \R$ be a real function.
Let $f$ be discontinuous at $a \in A$.
{{TFAE|def = Removable Discontinuity of Real Function|view = removable discontinuity}}
=== Definition 1 ===
{{Definition:Removable Discontinuity of Real Function/Definition 1}}
=== Definit... | === Lemma ===
{{:Equivalence of Definitions of Removable Discontinuity of Real Function/Lemma}}{{qed|lemma}}
Let $A \subseteq \R$ be a subset of the real numbers.
Let $f: A \to \R$ be a real function.
Let $f$ be discontinuous at $a \in A$.
For any $b \in \R$, define the function $f_b$ by:
:$\map {f_b} x = \begin {cases... | Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Let $f: A \to \R$ be a [[definition:Real Function|real function]].
Let $f$ be [[Definition:Discontinuous Real Function at Point|discontinuous]] at $a \in A$.
{{TFAE|def = Removable Discontinuity of Real Function|v... | === [[Equivalence of Definitions of Removable Discontinuity of Real Function/Lemma|Lemma]] ===
{{:Equivalence of Definitions of Removable Discontinuity of Real Function/Lemma}}{{qed|lemma}}
Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]].
Let $f: A \to \R$ be a ... | Equivalence of Definitions of Removable Discontinuity of Real Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Removable_Discontinuity_of_Real_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Removable_Discontinuity_of_Real_Function | [
"Removable Discontinuities (Real Analysis)",
"Equivalence of Definitions of Removable Discontinuity of Real Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"definition:Real Function",
"Definition:Discontinuous Mapping/Real Function/Point",
"Definition:Discontinuity (Real Analysis)/Removable/Definition 1",
"Definition:Discontinuity (Real Analysis)/Removable/Definition 2"
] | [
"Equivalence of Definitions of Removable Discontinuity of Real Function/Lemma",
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Discontinuous Mapping/Real Function/Point",
"Definition:Real Function",
"Equivalence of Definitions of Removable Discontinuity of Real Fu... |
proofwiki-13773 | Equivalence of Definitions of Polynomial Function on Subset of Ring | Let $R$ be a commutative ring with unity.
Let $S \subset R$ be a subset.
{{TFAE|def = Polynomial Function/Ring |view = polynomial function}} | === 1 implies 2 ===
Let $\map P X \in R \sqbrk X$ be the polynomial:
:$P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$
where $\sum$ denotes indexed summation.
We show that $\map P \iota = f$.
{{finish}} | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $S \subset R$ be a [[Definition:Subset|subset]].
{{TFAE|def = Polynomial Function/Ring |view = polynomial function}} | === 1 implies 2 ===
Let $\map P X \in R \sqbrk X$ be the [[Definition:Polynomial|polynomial]]:
:$P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$
where $\sum$ denotes [[Definition:Indexed Summation|indexed summation]].
We show that $\map P \iota = f$.
{{finish}} | Equivalence of Definitions of Polynomial Function on Subset of Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_Function_on_Subset_of_Ring | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_Function_on_Subset_of_Ring | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Subset"
] | [
"Definition:Polynomial",
"Definition:Summation/Indexed",
"Definition:Polynomial",
"Definition:Summation/Indexed"
] |
proofwiki-13774 | Monomials form Basis of Polynomial Ring/One Variable | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$.
Then the monomials of $R \sqbrk X$ are a basis of $R \sqbrk X$ as a module over $R$. | Follows from:
* Polynomial is Linear Combination of Monomials
* Monomials of Polynomial Ring are Linearly Independent
{{qed}}
Category:Polynomial Rings
Category:Monomials
ibj1ycvcvta6451o5b2hkeptfdnrhkb | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$ in the [[Definition:Variable of Polynomial Ring|variable]] $X$.
Then the [[Definition:Monomial of Polynomial Ring|monomials]] of $R \sqbrk X$ are a [[Defi... | Follows from:
* [[Polynomial is Linear Combination of Monomials]]
* [[Monomials of Polynomial Ring are Linearly Independent]]
{{qed}}
[[Category:Polynomial Rings]]
[[Category:Monomials]]
ibj1ycvcvta6451o5b2hkeptfdnrhkb | Monomials form Basis of Polynomial Ring/One Variable | https://proofwiki.org/wiki/Monomials_form_Basis_of_Polynomial_Ring/One_Variable | https://proofwiki.org/wiki/Monomials_form_Basis_of_Polynomial_Ring/One_Variable | [
"Polynomial Rings",
"Monomials"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Monomial of Polynomial Ring",
"Definition:Basis of Module",
"Definition:Module Structure of Polynomial Ring"
] | [
"Polynomial is Linear Combination of Monomials",
"Monomials of Polynomial Ring are Linearly Independent",
"Category:Polynomial Rings",
"Category:Monomials"
] |
proofwiki-13775 | Equivalence of Definitions of Polynomial in Ring Element | Let $R$ be a commutative ring.
Let $S$ be a subring with unity of $R$.
Let $x\in R$.
{{TFAE|def = Polynomial in Ring Element}} | {{proof wanted}}
Category:Polynomial Theory
imnf88w7v83rtn8p2kas888887ipv94 | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $S$ be a [[Definition:Subring|subring]] [[Definition:Ring with Unity|with unity]] of $R$.
Let $x\in R$.
{{TFAE|def = Polynomial in Ring Element}} | {{proof wanted}}
[[Category:Polynomial Theory]]
imnf88w7v83rtn8p2kas888887ipv94 | Equivalence of Definitions of Polynomial in Ring Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_in_Ring_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_in_Ring_Element | [
"Polynomial Theory"
] | [
"Definition:Commutative Ring",
"Definition:Subring",
"Definition:Ring with Unity"
] | [
"Category:Polynomial Theory"
] |
proofwiki-13776 | Union of Blocks is Set of Points | Let $\struct {X, \BB}$ be a pairwise balanced design.
That is, let $\struct {X, \BB}$ be a design, with $\size X \ge 2$, and the number of occurrences of each pair of distinct points in $\BB$ be $\lambda$ for some $\lambda > 0$ constant.
Then the set union of all the subset elements in $\BB$ is precisely $X$. | Let $X = \set {x_1, x_2, \ldots, x_v}$.
Let $\BB = \multiset {y_1, y_2,\ldots, y_b}$, where the notation denotes a multiset.
Let $Y = \ds \bigcup_{i \mathop = 1}^b y_i$.
We shall show that $Y \subseteq X$ and $X \subseteq Y$. | Let $\struct {X, \BB}$ be a [[Definition:Pairwise Balanced Design|pairwise balanced design]].
That is, let $\struct {X, \BB}$ be a [[Definition:Design|design]], with $\size X \ge 2$, and the number of occurrences of each pair of [[Definition:Distinct|distinct]] [[Definition:Point (Design Theory)|points]] in $\BB$ be $... | Let $X = \set {x_1, x_2, \ldots, x_v}$.
Let $\BB = \multiset {y_1, y_2,\ldots, y_b}$, where the notation denotes a [[Definition:Multiset|multiset]].
Let $Y = \ds \bigcup_{i \mathop = 1}^b y_i$.
We shall show that $Y \subseteq X$ and $X \subseteq Y$. | Union of Blocks is Set of Points | https://proofwiki.org/wiki/Union_of_Blocks_is_Set_of_Points | https://proofwiki.org/wiki/Union_of_Blocks_is_Set_of_Points | [
"Union of Blocks is Set of Points",
"Design Theory"
] | [
"Definition:Pairwise Balanced Design",
"Definition:Block Design",
"Definition:Distinct",
"Definition:Point (Design Theory)",
"Definition:Constant",
"Definition:Set Union",
"Definition:Subset",
"Definition:Element"
] | [
"Definition:Multiset",
"Definition:Multiset"
] |
proofwiki-13777 | Equality of Monomials of Polynomial Ring in One Variable | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.
Let $k, l \in \N$ be distinct natural numbers.
Then the mononomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$. | By:
* Uniqueness of Polynomial Ring in One Variable
* Homomorphism Preserves Indexed Products
we may assume $R \sqbrk X$ is the ring of sequences of finite support over $R$, and $X$ is the sequence $\sequence {0, 1, 0, 0 \ldots}$.
One verifies that, for $k \ge 0$, $X^k$ is the sequence with $\map {X^k} l = \delta_{k l}... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] $X$ over $R$.
Let $k, l \in \N$ be [[Definition:Distinct|distinct]] [[Definition:Natural Number|natural numbers]].
Then the [[Defi... | By:
* [[Uniqueness of Polynomial Ring in One Variable]]
* [[Homomorphism Preserves Indexed Products]]
we may assume $R \sqbrk X$ is the [[Definition:Ring of Sequences of Finite Support|ring of sequences of finite support]] over $R$, and $X$ is the [[Definition:Sequence|sequence]] $\sequence {0, 1, 0, 0 \ldots}$.
One v... | Equality of Monomials of Polynomial Ring in One Variable | https://proofwiki.org/wiki/Equality_of_Monomials_of_Polynomial_Ring_in_One_Variable | https://proofwiki.org/wiki/Equality_of_Monomials_of_Polynomial_Ring_in_One_Variable | [
"Monomials"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Distinct",
"Definition:Natural Numbers",
"Definition:Monomial of Polynomial Ring",
"Definition:Distinct",
"Definition:Power of Element/Ring"
] | [
"Uniqueness of Polynomial Ring in One Variable",
"Homomorphism Preserves Indexed Products",
"Definition:Ring of Sequences of Finite Support",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Kronecker Delta"
] |
proofwiki-13778 | Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.
Let $\iota : R \to R \sqbrk X$ denote the canonical embedding.
Let $S$ be a commutative ring with unity and $f: R \sqbrk X \to S$ be a ring isomorphism.
Then $\struct {S, f \circ \iota, \map f X}$ is a polynomi... | {{proof wanted}}
Category:Polynomial Rings
4awnnfe99b8800jwaxzarzp3xfiq024 | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] $X$ over $R$.
Let $\iota : R \to R \sqbrk X$ denote the [[Definition:Embedding into Polynomial Ring|canonical embedding]].
Let $S$ ... | {{proof wanted}}
[[Category:Polynomial Rings]]
4awnnfe99b8800jwaxzarzp3xfiq024 | Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable | https://proofwiki.org/wiki/Ring_Isomorphic_to_Polynomial_Ring_is_Polynomial_Ring/One_Variable | https://proofwiki.org/wiki/Ring_Isomorphic_to_Polynomial_Ring_is_Polynomial_Ring/One_Variable | [
"Polynomial Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Embedding",
"Definition:Commutative and Unitary Ring",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Indeterminate"
] | [
"Category:Polynomial Rings"
] |
proofwiki-13779 | Mittag-Leffler Expansion for Cosecant Function | :$\ds \pi \cosec \pi z = \frac 1 z + 2\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 - n^2}$ | {{begin-eqn}}
{{eqn | l = \pi \cosec \pi z
| r = \map \Gamma z \map \Gamma {1 - z}
| c = Euler's Reflection Formula
}}
{{eqn | r = \map \Beta {z, 1 - z}
| c = {{Defof|Beta Function|index = 3}}
}}
{{eqn | r = \int_{\mathop \to 0}^{\mathop \to 1} t^{z - 1} \paren {1 - t}^{-z} \rd t
| c = {{Defof|B... | :$\ds \pi \cosec \pi z = \frac 1 z + 2\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 - n^2}$ | {{begin-eqn}}
{{eqn | l = \pi \cosec \pi z
| r = \map \Gamma z \map \Gamma {1 - z}
| c = [[Euler's Reflection Formula]]
}}
{{eqn | r = \map \Beta {z, 1 - z}
| c = {{Defof|Beta Function|index = 3}}
}}
{{eqn | r = \int_{\mathop \to 0}^{\mathop \to 1} t^{z - 1} \paren {1 - t}^{-z} \rd t
| c = {{Def... | Mittag-Leffler Expansion for Cosecant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cosecant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cosecant_Function | [
"Mittag-Leffler Expansion for Cosecant Function",
"Cosecant Function",
"Mittag-Leffler Expansions"
] | [] | [
"Euler's Reflection Formula",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Reversal of Limits of Definite Integral",
"Power Series Expansion for Reciprocal of 1 + x",
"Integral of Power",
"Difference of Two Squares"
] |
proofwiki-13780 | Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions | Let $\closedint a b$ be a closed real interval.
Let $f$ be a bounded real function defined on $\closedint a b$.
Let $P$ and $Q$ be finite subdivisions of $\closedint a b$.
Let $\map L P$ be the lower Darboux sum of $f$ on $\closedint a b$ with respect to $P$.
Let $\map U Q$ be the upper Darboux sum of $f$ on $\closedin... | Let $P' = P \cup Q$.
We observe:
:$P'$ is either equal to $P$ or finer than $P$
:$P'$ is either equal to $Q$ or finer than $Q$
We find:
:$\map L P \le \map L {P'}$ by Lower Sum of Refinement
:$\map L {P'} \le \map U {P'}$ by Upper Darboux Sum Never Smaller than Lower Darboux Sum
:$\map U {P'} \le \map U Q$ by Upper Sum... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real function]] defined on $\closedint a b$.
Let $P$ and $Q$ be [[Definition:Finite Subdivision|finite subdivisions]] of $\closedint a b$.
Le... | Let $P' = P \cup Q$.
We observe:
:$P'$ is either equal to $P$ or [[Definition:Refinement of Finite Subdivision|finer]] than $P$
:$P'$ is either equal to $Q$ or [[Definition:Refinement of Finite Subdivision|finer]] than $Q$
We find:
:$\map L P \le \map L {P'}$ by [[Lower Sum of Refinement]]
:$\map L {P'} \le \map... | Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions | https://proofwiki.org/wiki/Upper_Sum_Never_Smaller_than_Lower_Sum_for_any_Pair_of_Subdivisions | https://proofwiki.org/wiki/Upper_Sum_Never_Smaller_than_Lower_Sum_for_any_Pair_of_Subdivisions | [
"Real Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Subdivision of Interval/Finite",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum"
] | [
"Definition:Refinement of Finite Subdivision",
"Definition:Refinement of Finite Subdivision",
"Lower Sum of Refinement",
"Upper Darboux Sum Never Smaller than Lower Darboux Sum",
"Upper Sum of Refinement"
] |
proofwiki-13781 | Set Finite iff Surjection from Initial Segment of Natural Numbers | Let $S$ be a set.
Then $S$ is finite {{iff}} for some $n \in \N$ there exists a surjection $f: \N_{<n} \to S$.
Here, $\N_{<n}$ denotes an initial segment of $\N$. | === Necessary Condition ===
Suppose that $S$ is finite.
By definition, this means there exists a bijection $f: \N_{<n} \to S$.
Then $f$ is a fortiori also the sought surjection.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Then $S$ is [[Definition:Finite Set|finite]] {{iff}} for some $n \in \N$ there exists a [[Definition:Surjection|surjection]] $f: \N_{<n} \to S$.
Here, $\N_{<n}$ denotes an [[Definition:Initial Segment of Natural Numbers|initial segment of $\N$]]. | === Necessary Condition ===
Suppose that $S$ is [[Definition:Finite Set|finite]].
By definition, this means there exists a [[Definition:Bijection|bijection]] $f: \N_{<n} \to S$.
Then $f$ is [[Definition:A Fortiori|a fortiori]] also the sought [[Definition:Surjection|surjection]].
{{qed|lemma}} | Set Finite iff Surjection from Initial Segment of Natural Numbers | https://proofwiki.org/wiki/Set_Finite_iff_Surjection_from_Initial_Segment_of_Natural_Numbers | https://proofwiki.org/wiki/Set_Finite_iff_Surjection_from_Initial_Segment_of_Natural_Numbers | [
"Set Theory",
"Surjections"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Surjection",
"Definition:Initial Segment of Natural Numbers"
] | [
"Definition:Finite Set",
"Definition:Bijection",
"Definition:A Fortiori",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Finite Set"
] |
proofwiki-13782 | Set Finite iff Injection to Initial Segment of Natural Numbers | Let $S$ be a set.
Then $S$ is finite {{iff}} for some $n \in \N$ there exists an injection $f: S \to \N_{< n}$.
Here, $\N_{< n}$ denotes an initial segment of $\N$. | === Necessary Condition ===
Suppose that $S$ is finite.
By definition, this means there exists a bijection $f: S \to \N_{< n}$.
Then $f$ is a fortiori also the sought injection.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Then $S$ is [[Definition:Finite Set|finite]] {{iff}} for some $n \in \N$ there exists an [[Definition:Injection|injection]] $f: S \to \N_{< n}$.
Here, $\N_{< n}$ denotes an [[Definition:Initial Segment of Natural Numbers|initial segment of $\N$]]. | === Necessary Condition ===
Suppose that $S$ is [[Definition:Finite Set|finite]].
By definition, this means there exists a [[Definition:Bijection|bijection]] $f: S \to \N_{< n}$.
Then $f$ is [[Definition:A Fortiori|a fortiori]] also the sought [[Definition:Injection|injection]].
{{qed|lemma}} | Set Finite iff Injection to Initial Segment of Natural Numbers | https://proofwiki.org/wiki/Set_Finite_iff_Injection_to_Initial_Segment_of_Natural_Numbers | https://proofwiki.org/wiki/Set_Finite_iff_Injection_to_Initial_Segment_of_Natural_Numbers | [
"Set Theory",
"Injections"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Injection",
"Definition:Initial Segment of Natural Numbers"
] | [
"Definition:Finite Set",
"Definition:Bijection",
"Definition:A Fortiori",
"Definition:Injection",
"Definition:Injection",
"Definition:Finite Set"
] |
proofwiki-13783 | Largest Integer whose Digits taken in Pairs all form Distinct Primes | The largest integer which has the property that every pair of its digits taken together is a distinct prime number is $619 \, 737 \, 131 \, 179$. | Let $p$ be such an integer.
Apart from the first digit, each of its digits forms the second digit of a $2$-digit prime number.
Thus, apart from the first digit, $p$ cannot contain $2$, $4$, $5$, $6$, $8$ or $0$.
$0$ cannot of course be the first digit either.
Thus, apart from the first pair of its digits, the $2$-digit... | The largest [[Definition:Integer|integer]] which has the property that every pair of its [[Definition:Digit|digits]] taken together is a [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime number]] is $619 \, 737 \, 131 \, 179$. | Let $p$ be such an [[Definition:Integer|integer]].
Apart from the first [[Definition:Digit|digit]], each of its [[Definition:Digit|digits]] forms the second [[Definition:Digit|digit]] of a $2$-[[Definition:Digit|digit]] [[Definition:Prime Number|prime number]].
Thus, apart from the first [[Definition:Digit|digit]], $... | Largest Integer whose Digits taken in Pairs all form Distinct Primes | https://proofwiki.org/wiki/Largest_Integer_whose_Digits_taken_in_Pairs_all_form_Distinct_Primes | https://proofwiki.org/wiki/Largest_Integer_whose_Digits_taken_in_Pairs_all_form_Distinct_Primes | [
"Prime Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Distinct",
"Definition:Prime Number"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Set",
"Definition:Element",
"Definition:El... |
proofwiki-13784 | Finite Product of Finite Sets is Finite | Let $\sequence {S_n}$ be a sequence of finite sets.
Let $\ds \prod_{k \mathop = 1}^n S_k$ be their Cartesian product.
Then $\ds \prod_{k \mathop = 1}^n S_k$ is also a finite set. | {{ProofWanted|Boring induction from Product of Finite Sets is Finite}} | Let $\sequence {S_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Finite Set|finite sets]].
Let $\ds \prod_{k \mathop = 1}^n S_k$ be their [[Definition:Finite Cartesian Product|Cartesian product]].
Then $\ds \prod_{k \mathop = 1}^n S_k$ is also a [[Definition:Finite Set|finite set]]. | {{ProofWanted|Boring induction from [[Product of Finite Sets is Finite]]}} | Finite Product of Finite Sets is Finite | https://proofwiki.org/wiki/Finite_Product_of_Finite_Sets_is_Finite | https://proofwiki.org/wiki/Finite_Product_of_Finite_Sets_is_Finite | [
"Cartesian Product"
] | [
"Definition:Sequence",
"Definition:Finite Set",
"Definition:Cartesian Product/Finite",
"Definition:Finite Set"
] | [
"Product of Finite Sets is Finite"
] |
proofwiki-13785 | Smallest 22 Primes in Arithmetic Sequence | The smallest $22$ primes in arithmetic sequence are:
:$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$
for $n = 0, 1, \ldots, 21$. | {{begin-eqn}}
{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 0 \times 4 \, 609 \, 098 \, 694 \, 200
| r = 11 \, 410 \, 337 \, 850 \, 553
| c = which is prime
}}
{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 1 \times 4 \, 609 \, 098 \, 694 \, 200
| r = 16 \, 019 \, 436 \, 544 \, 753
| c = which is p... | The smallest $22$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
:$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$
for $n = 0, 1, \ldots, 21$. | {{begin-eqn}}
{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 0 \times 4 \, 609 \, 098 \, 694 \, 200
| r = 11 \, 410 \, 337 \, 850 \, 553
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 11 \, 410 \, 337 \, 850 \, 553 + 1 \times 4 \, 609 \, 098 \, 694 \, 200
| r = 16 \, 019 \, 436 \, 544 \... | Smallest 22 Primes in Arithmetic Sequence | https://proofwiki.org/wiki/Smallest_22_Primes_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Smallest_22_Primes_in_Arithmetic_Sequence | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Prime Number",
"Definition:Arithmetic Sequence"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",... |
proofwiki-13786 | Smallest Strong Fibonacci Pseudoprime of Type I | The smallest strong Fibonacci pseudoprime of type I is $443 \, 372 \, 888 \, 629 \, 441$. | Let $N := 443 \, 372 \, 888 \, 629 \, 441$, to save writing it out in full each time.
From the definition of a strong Fibonacci pseudoprime of type I:
{{:Definition:Strong Fibonacci Pseudoprime of Type I}}
We have that:
:$N = 17 \times 31 \times 41 \times 43 \times 89 \times 97 \times 167 \times 331$
Of these, we see t... | The smallest [[Definition:Strong Fibonacci Pseudoprime of Type I|strong Fibonacci pseudoprime of type I]] is $443 \, 372 \, 888 \, 629 \, 441$. | Let $N := 443 \, 372 \, 888 \, 629 \, 441$, to save writing it out in full each time.
From the definition of a [[Definition:Strong Fibonacci Pseudoprime of Type I|strong Fibonacci pseudoprime of type I]]:
{{:Definition:Strong Fibonacci Pseudoprime of Type I}}
We have that:
:$N = 17 \times 31 \times 41 \times 43 \time... | Smallest Strong Fibonacci Pseudoprime of Type I | https://proofwiki.org/wiki/Smallest_Strong_Fibonacci_Pseudoprime_of_Type_I | https://proofwiki.org/wiki/Smallest_Strong_Fibonacci_Pseudoprime_of_Type_I | [
"Strong Fibonacci Pseudoprimes",
"443,372,888,629,441"
] | [
"Definition:Strong Fibonacci Pseudoprime/Type I"
] | [
"Definition:Strong Fibonacci Pseudoprime/Type I",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Even Integer",
"De... |
proofwiki-13787 | Infimum of Upper Sums Never Smaller than Lower Sum | :$\inf_P \map U P \ge \map L S$ | From Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions, $\map L S$ is a lower bound for the real set:
:$T = \leftset {\map U P: P}$ is a finite subdivision of $\rightset {\closedint a b}$
Since $\inf_P \map U P$ is the infumum of $T$:
:$\inf_P \map U P \ge \map L S$
Hence the result.
{{qed}}
Category:... | :$\inf_P \map U P \ge \map L S$ | From [[Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions]], $\map L S$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for the [[Definition:Real Number|real]] [[Definition:Set|set]]:
:$T = \leftset {\map U P: P}$ is a [[Definition:Finite Subdivision|finite subdivision]] of $\righ... | Infimum of Upper Sums Never Smaller than Lower Sum | https://proofwiki.org/wiki/Infimum_of_Upper_Sums_Never_Smaller_than_Lower_Sum | https://proofwiki.org/wiki/Infimum_of_Upper_Sums_Never_Smaller_than_Lower_Sum | [
"Real Analysis"
] | [] | [
"Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Real Number",
"Definition:Set",
"Definition:Subdivision of Interval/Finite",
"Definition:Infimum of Set/Real Numbers",
"Category:Real Analysis"
] |
proofwiki-13788 | Supremum of Lower Sums Never Greater than Upper Sum | :$\sup_P \map L P \le \map U S$ | From Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions, $\map U S$ is an upper bound for the real set:
:$T = \leftset {\map L P: P}$ is a finite subdivision of $\rightset {\closedint a b}$
Since $\sup_P \map L P$ is the supremum of $T$:
:$\sup_P \map L P \le \map U S$
Hence the result.
{{qed}}
Categor... | :$\sup_P \map L P \le \map U S$ | From [[Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions]], $\map U S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for the [[Definition:Real Number|real]] [[Definition:Set|set]]:
:$T = \leftset {\map L P: P}$ is a [[Definition:Finite Subdivision|finite subdivision]] of $\rig... | Supremum of Lower Sums Never Greater than Upper Sum | https://proofwiki.org/wiki/Supremum_of_Lower_Sums_Never_Greater_than_Upper_Sum | https://proofwiki.org/wiki/Supremum_of_Lower_Sums_Never_Greater_than_Upper_Sum | [
"Real Analysis"
] | [] | [
"Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Real Number",
"Definition:Set",
"Definition:Subdivision of Interval/Finite",
"Definition:Supremum of Set/Real Numbers",
"Category:Real Analysis"
] |
proofwiki-13789 | Smallest Cunningham Chain of the Second Kind of Length 13 | The smallest Cunningham chain of the second kind of length $13$ is:
:$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 \, 343 \, 165 \, 711 \, 201$, $24 \, 258 \, 686 \, 331 \, 422 \, 401$, $48 \, 517 \, ... | Let $C$ denote the sequence in question.
We have that $758 \, 083 \, 947 \, 856 \, 951$ is prime.
First note that:
:$\dfrac {758 \, 083 \, 947 \, 856 \, 951 + 1} 2 = 379 \, 041 \, 973 \, 928 \, 476 = 2^2 \times 94 \, 760 \, 493 \, 482 \, 119$
and so is not prime.
Thus $758 \, 083 \, 947 \, 856 \, 951$ fulfils the requi... | The smallest [[Definition:Cunningham Chain of the Second Kind|Cunningham chain of the second kind]] of [[Definition:Length of Sequence|length]] $13$ is:
:$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 ... | Let $C$ denote the [[Definition:Sequence|sequence]] in question.
We have that $758 \, 083 \, 947 \, 856 \, 951$ is [[Definition:Prime Number|prime]].
First note that:
:$\dfrac {758 \, 083 \, 947 \, 856 \, 951 + 1} 2 = 379 \, 041 \, 973 \, 928 \, 476 = 2^2 \times 94 \, 760 \, 493 \, 482 \, 119$
and so is not [[Defini... | Smallest Cunningham Chain of the Second Kind of Length 13 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_Second_Kind_of_Length_13 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_Second_Kind_of_Length_13 | [
"Cunningham Chains"
] | [
"Definition:Cunningham Chain/Second Kind",
"Definition:Length of Sequence"
] | [
"Definition:Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Cunningham Chain/Second Kind",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Pr... |
proofwiki-13790 | Properties of 5,559,060,566,555,523 | $3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties:
: It has a remarkably large number of $5$s (half of its digits).
: Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular digit.
: Multiply it by $8$, and $9$ of the digits of the result are $4$. | {{begin-eqn}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 2
| r = 11 \, 118 \, 121 \, 133 \, 111 \, 046
| c = $10$ $1$s
}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 4
| r = 22 \, 236 \, 242 \, 266 \, 222 \, 092
| c = $10$ $2$s
}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 55... | $3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties:
: It has a remarkably large number of $5$s (half of its [[Definition:Digit|digits]]).
: Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular [[Definition:Digit|digit]].
: Multiply it by $8$, and $9$ of the [[Definition:D... | {{begin-eqn}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 2
| r = 11 \, 118 \, 121 \, 133 \, 111 \, 046
| c = $10$ $1$s
}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 4
| r = 22 \, 236 \, 242 \, 266 \, 222 \, 092
| c = $10$ $2$s
}}
{{eqn | l = 5 \, 559 \, 060 \, 566 \, 55... | Properties of 5,559,060,566,555,523 | https://proofwiki.org/wiki/Properties_of_5,559,060,566,555,523 | https://proofwiki.org/wiki/Properties_of_5,559,060,566,555,523 | [
"5,559,060,566,555,523",
"Recreational Mathematics"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Digit"
] | [] |
proofwiki-13791 | First Occurrence of Prime Gap of 864 | The first occurrence of a prime gap of $864$ is between $6 \, 505 \, 941 \, 701 \, 960 \, 039$ and $6 \, 505 \, 941 \, 701 \, 960 \, 903$. | $6 \, 505 \, 941 \, 701 \, 960 \, 039$ is a prime number.
$6 \, 505 \, 941 \, 701 \, 960 \, 903$ is a prime number.
It can be checked that all numbers between these two are composite.
It can also be checked by computer that there is no prime gap of $864$ between smaller prime numbers.
{{qed}} | The first occurrence of a [[Definition:Prime Gap|prime gap]] of $864$ is between $6 \, 505 \, 941 \, 701 \, 960 \, 039$ and $6 \, 505 \, 941 \, 701 \, 960 \, 903$. | $6 \, 505 \, 941 \, 701 \, 960 \, 039$ is a [[Definition:Prime Number|prime number]].
$6 \, 505 \, 941 \, 701 \, 960 \, 903$ is a [[Definition:Prime Number|prime number]].
It can be checked that all numbers between these two are [[Definition:Composite Number|composite]].
It can also be checked by computer that there... | First Occurrence of Prime Gap of 864 | https://proofwiki.org/wiki/First_Occurrence_of_Prime_Gap_of_864 | https://proofwiki.org/wiki/First_Occurrence_of_Prime_Gap_of_864 | [
"Prime Gaps"
] | [
"Definition:Prime Gap"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Gap",
"Definition:Prime Number"
] |
proofwiki-13792 | Reciprocal of 19 from Sum of Powers of 2 Backwards | The decimal expansion of the reciprocal of $19$ can be constructed by summing the powers of $2$, offset progressively backwards by $1$ digit:
<pre>
1
2
4
8
16
32
64
... | We confirm that from Reciprocal of $19$:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
The construction above can be expressed as the sum:
:$\ds \sum_{k \mathop \ge 0} \paren {2^k \times 10^k} = \sum_{k \mathop \ge 0} 20^k$
Obviously this sum does not converge.
However we can still analyse the l... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $19$ can be constructed by [[Definition:Integer Addition|summing]] the [[Definition:Integer Power|powers]] of $2$, offset progressively backwards by $1$ digit:
<pre>
1
... | We confirm that from [[Reciprocal of 19|Reciprocal of $19$]]:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
The construction above can be expressed as the [[Definition:Integer Addition|sum]]:
:$\ds \sum_{k \mathop \ge 0} \paren {2^k \times 10^k} = \sum_{k \mathop \ge 0} 20^k$
Obviously this [... | Reciprocal of 19 from Sum of Powers of 2 Backwards | https://proofwiki.org/wiki/Reciprocal_of_19_from_Sum_of_Powers_of_2_Backwards | https://proofwiki.org/wiki/Reciprocal_of_19_from_Sum_of_Powers_of_2_Backwards | [
"19",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer"
] | [
"Reciprocal of 19",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Convergent Series/Number Field",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Number times Recurring Part of Reciprocal gives 9-Repdigit/Generalization",
"Definition:Digit",
"... |
proofwiki-13793 | General Fibonacci Sequence whose Terms are all Composite | The general Fibonacci sequence $\left\langle{a_n}\right\rangle$ defined as:
:<nowiki>$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
a_{n - 2} + a_{n - 1} & : n > 1
\end{cases}$</nowiki>
where:
:$r = 62 \, 638 \, 280 \, 004 \, 239 \, 857$
:$s = 49 \, 463 \, 435 \, 743 \, 205 \, 655$
is such that:
: $r$ and $s$ are c... | {{ProofWanted|Knuth proves it in the citation given. Needs to be transcribed, but it's boring and I can't be bothered.}} | The [[Definition:General Fibonacci Sequence|general Fibonacci sequence]] $\left\langle{a_n}\right\rangle$ defined as:
:<nowiki>$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
a_{n - 2} + a_{n - 1} & : n > 1
\end{cases}$</nowiki>
where:
:$r = 62 \, 638 \, 280 \, 004 \, 239 \, 857$
:$s = 49 \, 463 \, 435 \, 743 \, 20... | {{ProofWanted|Knuth proves it in the citation given. Needs to be transcribed, but it's boring and I can't be bothered.}} | General Fibonacci Sequence whose Terms are all Composite | https://proofwiki.org/wiki/General_Fibonacci_Sequence_whose_Terms_are_all_Composite | https://proofwiki.org/wiki/General_Fibonacci_Sequence_whose_Terms_are_all_Composite | [
"Fibonacci Numbers"
] | [
"Definition:General Fibonacci Sequence",
"Definition:Coprime/Integers",
"Definition:Term of Sequence",
"Definition:Composite Number"
] | [] |
proofwiki-13794 | Superset of Infinite Set is Infinite | Let $S$ be an infinite set.
Let $T \supseteq S$ be a superset of $S$.
Then $T$ is also infinite. | Suppose $T$ were finite.
Then by Set Finite iff Injection to Initial Segment of Natural Numbers, there is an injection $f: T \to \N_{<n}$ for some $n \in \N$.
But then by Restriction of Injection is Injection, also the restriction of $f$ to $S$:
:$f {\restriction_S}: S \to \N_{<n}$
is an injection.
Again by Set Finite ... | Let $S$ be an [[Definition:Infinite Set|infinite set]].
Let $T \supseteq S$ be a [[Definition:Superset|superset]] of $S$.
Then $T$ is also [[Definition:Infinite Set|infinite]]. | Suppose $T$ were [[Definition:Finite Set|finite]].
Then by [[Set Finite iff Injection to Initial Segment of Natural Numbers]], there is an [[Definition:Injection|injection]] $f: T \to \N_{<n}$ for some $n \in \N$.
But then by [[Restriction of Injection is Injection]], also the [[Definition:Restriction of Mapping|rest... | Superset of Infinite Set is Infinite | https://proofwiki.org/wiki/Superset_of_Infinite_Set_is_Infinite | https://proofwiki.org/wiki/Superset_of_Infinite_Set_is_Infinite | [
"Set Theory"
] | [
"Definition:Infinite Set",
"Definition:Subset/Superset",
"Definition:Infinite Set"
] | [
"Definition:Finite Set",
"Set Finite iff Injection to Initial Segment of Natural Numbers",
"Definition:Injection",
"Restriction of Injection is Injection",
"Definition:Restriction/Mapping",
"Definition:Injection",
"Set Finite iff Injection to Initial Segment of Natural Numbers",
"Definition:Infinite S... |
proofwiki-13795 | Prime Decomposition of 2^58+1 | The number $2^{58} + 1$ has the prime decomposition:
:$2^{58} + 1 = 5 \times 107 \, 367 \, 629 \times 536 \, 903 \, 681$ | From Aurifeuillian Factorization of 2 Mod 4th Power of Two plus 1, we have:
:$2^{4 n + 2} + 1 = \paren {2^{2 n + 1} - 2^{n + 1} + 1} \paren {2^{2 n + 1} + 2^{n + 1} + 1}$
Setting $n = 14$:
{{begin-eqn}}
{{eqn | l = 2^{58} + 1
| r = \paren {2^{29} - 2^{15} + 1} \paren {2^{29} + 2^{15} + 1}
| c =
}}
{{eqn | ... | The number $2^{58} + 1$ has the [[Definition:Prime Decomposition|prime decomposition]]:
:$2^{58} + 1 = 5 \times 107 \, 367 \, 629 \times 536 \, 903 \, 681$ | From [[Aurifeuillian Factorization of 2 Mod 4th Power of Two plus 1]], we have:
:$2^{4 n + 2} + 1 = \paren {2^{2 n + 1} - 2^{n + 1} + 1} \paren {2^{2 n + 1} + 2^{n + 1} + 1}$
Setting $n = 14$:
{{begin-eqn}}
{{eqn | l = 2^{58} + 1
| r = \paren {2^{29} - 2^{15} + 1} \paren {2^{29} + 2^{15} + 1}
| c =
}}
{... | Prime Decomposition of 2^58+1 | https://proofwiki.org/wiki/Prime_Decomposition_of_2^58+1 | https://proofwiki.org/wiki/Prime_Decomposition_of_2^58+1 | [
"Prime Numbers",
"288,230,376,151,711,745"
] | [
"Definition:Prime Decomposition"
] | [
"Aurifeuillian Factorization/Examples/2^4n+2 + 1"
] |
proofwiki-13796 | Sum of Squares as Product of Factors with Square Roots | :$x^2 + y^2 = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}$ | {{begin-eqn}}
{{eqn | l = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}
| r = x^2 - x \sqrt {2 x y} + x y + x \sqrt {2 x y} - \sqrt {2 x y} \sqrt {2 x y} + y \sqrt {2 x y} + x y - y \sqrt {2 x y} + y^2
| c =
}}
{{eqn | r = x^2 + \paren {x - x} \sqrt {2 x y} + 2 x y - 2 x y + \paren {y - y} ... | :$x^2 + y^2 = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}$ | {{begin-eqn}}
{{eqn | l = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}
| r = x^2 - x \sqrt {2 x y} + x y + x \sqrt {2 x y} - \sqrt {2 x y} \sqrt {2 x y} + y \sqrt {2 x y} + x y - y \sqrt {2 x y} + y^2
| c =
}}
{{eqn | r = x^2 + \paren {x - x} \sqrt {2 x y} + 2 x y - 2 x y + \paren {y - y} ... | Sum of Squares as Product of Factors with Square Roots | https://proofwiki.org/wiki/Sum_of_Squares_as_Product_of_Factors_with_Square_Roots | https://proofwiki.org/wiki/Sum_of_Squares_as_Product_of_Factors_with_Square_Roots | [
"Algebra",
"Sums of Squares"
] | [] | [
"Category:Algebra",
"Category:Sums of Squares"
] |
proofwiki-13797 | Repunit Expressed using Power of 10 | The repunit number $R_n$ can be expressed as:
:$R_n = \dfrac {10^n - 1} 9$ | {{begin-eqn}}
{{eqn | l = \dfrac {10^n - 1} 9
| r = \dfrac {10^n - 1} {10 - 1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 10^k
| c = Sum of Geometric Sequence
}}
{{eqn | r = 1 + 10 + 100 + \ldots + 10^{n - 2} + 10^{n - 1}
| c =
}}
{{end-eqn}}
The result follows from the Basis Representa... | The [[Definition:Repunit|repunit number]] $R_n$ can be expressed as:
:$R_n = \dfrac {10^n - 1} 9$ | {{begin-eqn}}
{{eqn | l = \dfrac {10^n - 1} 9
| r = \dfrac {10^n - 1} {10 - 1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^{n - 1} 10^k
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | r = 1 + 10 + 100 + \ldots + 10^{n - 2} + 10^{n - 1}
| c =
}}
{{end-eqn}}
The result follows from the [[Basis Rep... | Repunit Expressed using Power of 10 | https://proofwiki.org/wiki/Repunit_Expressed_using_Power_of_10 | https://proofwiki.org/wiki/Repunit_Expressed_using_Power_of_10 | [
"Repunits"
] | [
"Definition:Repunit"
] | [
"Sum of Geometric Sequence",
"Basis Representation Theorem"
] |
proofwiki-13798 | Repunit cannot be Square | A repunit (apart from the trivial $1$) cannot be a square. | Let $m$ be a repunit with $r$ digits such that $r > 1$.
By definition, $m$ is odd.
Thus from Square Modulo 4, if $m$ were square it would be of the form:
:$m \equiv 1 \pmod 4$.
$m$ is of the form $\ds \sum_{k \mathop = 0}^{r - 1} 10^k$ where $r$ is the number of digits.
Thus for $r \ge 2$:
{{begin-eqn}}
{{eqn | l = m
... | A [[Definition:Repunit|repunit]] (apart from the trivial $1$) cannot be a [[Definition:Square Number|square]]. | Let $m$ be a [[Definition:Repunit|repunit]] with $r$ [[Definition:Digit|digits]] such that $r > 1$.
By definition, $m$ is [[Definition:Odd Integer|odd]].
Thus from [[Square Modulo 4]], if $m$ were [[Definition:Square Number|square]] it would be of the form:
:$m \equiv 1 \pmod 4$.
$m$ is of the form $\ds \sum_{k \ma... | Repunit cannot be Square | https://proofwiki.org/wiki/Repunit_cannot_be_Square | https://proofwiki.org/wiki/Repunit_cannot_be_Square | [
"Repunits",
"Square Numbers"
] | [
"Definition:Repunit",
"Definition:Square Number"
] | [
"Definition:Repunit",
"Definition:Digit",
"Definition:Odd Integer",
"Square Modulo 4",
"Definition:Square Number",
"Definition:Digit",
"Definition:Square Number"
] |
proofwiki-13799 | Second Principle of Recursive Definition | Let $\N$ be the natural numbers.
Let $T$ be a set.
Let $a \in T$.
For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a mapping.
Then there exists exactly one mapping $f: \N \to T$ such that:
:$\forall x \in \N: \map f x = \begin{cases} a & : x = 0 \\ \map {G_n} {\map f 0, \ldots, \map f n} & : x = n + 1 \end{cases}$ | Define $T^*$ to be the Kleene closure of $T$:
:$T^* := \ds \bigcup_{i \mathop = 1}^\infty T^i$
Note that, for convenience, the empty sequence is excluded from $T^*$.
Now define a mapping $\GG: T^* \to T^*$ by:
:$\map \GG {t_1, \ldots, t_n} = \tuple {t_1, \ldots, t_n, \map {G_n} {t_1, \ldots, t_n} }$
that is, extending ... | Let $\N$ be the [[Definition:Natural Number|natural numbers]].
Let $T$ be a [[Definition:Set|set]].
Let $a \in T$.
For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a [[Definition:Mapping|mapping]].
Then there exists exactly one [[Definition:Mapping|mapping]] $f: \N \to T$ such that:
:$\forall x \in \N: \map f x ... | Define $T^*$ to be the [[Definition:Kleene Closure|Kleene closure]] of $T$:
:$T^* := \ds \bigcup_{i \mathop = 1}^\infty T^i$
Note that, for convenience, the [[Definition:Empty Sequence|empty sequence]] is excluded from $T^*$.
Now define a [[Definition:Mapping|mapping]] $\GG: T^* \to T^*$ by:
:$\map \GG {t_1, \ldots... | Second Principle of Recursive Definition | https://proofwiki.org/wiki/Second_Principle_of_Recursive_Definition | https://proofwiki.org/wiki/Second_Principle_of_Recursive_Definition | [
"Named Theorems",
"Mapping Theory",
"Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping"
] | [
"Definition:Kleene Closure",
"Definition:Sequence/Empty Sequence",
"Definition:Mapping",
"Definition:Finite Sequence",
"Definition:Element",
"Principle of Recursive Definition",
"Definition:Finite Sequence",
"Definition:Unique",
"Definition:Mapping",
"Principle of Mathematical Induction",
"Princ... |
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