hcju/proofwikips huggingface dataset:
Equivalence of Definitions of Polynomial Ring in One Variable
Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As a Ring of Sequences === {{:Definition:Polynomial Ring/Sequences}} === Definition 2: As a Monoid Ring on the Natural Numbers === {{:Definition:Polynomial Ring/Monoid Ring on Natural Numbers}}
Smallest Triplet of Integers whose Product with Tau Value are Equal
Let $\map \tau n$ denote the $\tau$ function: the number of divisors of $n$. The smallest set of $3$ integers $T$ such that $m \, \map \tau m$ is equal for each $m \in T$ is: :$\set {168, 192, 224}$
General Variation of Integral Functional/Dependent on n Variables
Let $\mathbf x$ be an $n$-dimensional vector. Let $u = \map u {\mathbf x}$ be a real-valued function. Let $J$ be a functional such that: :$\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$ Let $\mathbf x^*, u^*$ be defined by the following transformations $\paren \star$: {{begin-eqn}} {{eqn | l = \mathbf x^*       | r = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } + \map \OO {\epsilon^2} }} {{eqn | l = u^*       | r = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x } } + \map \OO {\epsilon^2} }} {{end-eqn}} where: :$\boldsymbol \Phi$, $\Psi$ are differentiable {{WRT|Differentiation}} $\epsilon$ and: {{begin-eqn}} {{eqn | l = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}       | r = \mathbf x }} {{eqn | l = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}       | r = u }} {{eqn | l = \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }       | r = \valueat {\dfrac {\partial \boldsymbol \Phi} {\partial \epsilon} } {\epsilon \mathop = 0} }} {{eqn | l = \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }       | r = \valueat {\dfrac {\partial \Psi} {\partial \epsilon} } {\epsilon \mathop = 0} }} {{end-eqn}} Then the variation of the functional $J$ due to the original mapping being transformed by $\paren \star$ reads: {{explain|not clear what "aforementioned transformation" refers to. Best to identify it with a label and refer to that label.}} :$\ds \delta J = \epsilon \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline \psi \rd x_1 \dotsm \rd x_n + \epsilon \int_R \map {\dfrac {\partial} {\partial \mathbf x} } {F_{u_x} \overline {\boldsymbol \psi} + F \boldsymbol \phi} \rd x_1 \dotsm \rd x_n$ where: :$\overline \psi = \psi - u_{\mathbf x} \boldsymbol \phi$
Order Isomorphism forms Galois Connection
Let $L_1 = \left({S_1, \preceq_1}\right)$, $L_2 = \left({S_2, \preceq_2}\right)$ be ordered sets. Let $f:S_1 \to S_2$ be an order isomorphism between $L_1$ and $L_2$. Then $\left({f, f^{-1} }\right)$ is a Galois connection.
Numbers of Primes with at most n Digits
Let $p: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$\forall n \in \Z_{>0}: p \left({n}\right) = $ the number of prime numbers with no more than $n$ digits Then the value of $p$ for the first few numbers is given below: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $p \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $25$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $168$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $1229$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $9592$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $78 \, 498$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $664 \, 579$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $5 \, 761 \, 455$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $50 \, 847 \, 534$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$ |} {{OEIS|A006880}}
Sequence of Square Centered Hexagonal Numbers
The sequence of centered hexagonal numbers which are also square begins: :$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$ {{OEIS|A006051}}
169 as Sum of up to 155 Squares
$169$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$.
Lattice of Power Set is Arithmetic
Let $X$ be a set. Let $P = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a lattice of power set. Then $P$ is arithmetic.
Nth Derivative of Natural Logarithm
The $n$th derivative of $\map \ln x$ for $n \ge 1$ is: :$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$
Sum of Factorials of Digits of 169
Let the factorials of the digits of $169$ be added. Let the same process be done on the result. Repeat. After $3$ iterations, the result will be $169$.
Squares which are Difference between Two Cubes
$169$ is the smallest square number which is the difference between two cubes: :$169 = 8^3 - 7^3$ {{expand|Add the rest of the sequence, having found out what they are.}}
Cube of 180 is Sum of Sequence of Consecutive Cubes
:$180^3 = \displaystyle \sum_{k \mathop = 6}^{69} k^3$ That is: :$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$
Numbers Expressible as Sum of Five Distinct Squares
The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$. Both $188$ and $124$ require as many as $6$ distinct squares to represent them: {{begin-eqn}} {{eqn | l = 124       | r = 1 + 4 + 9 + 25 + 36 + 49       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 188       | r = 1 + 4 + 9 + 25 + 49 + 100       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 10^2       | c =  }} {{end-eqn}}
Closed Form for Heptagonal Pyramidal Numbers
The closed-form expression for the $n$th heptagonal pyramidal number is: :$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$
Heptagonal Pyramidal Numbers which are Square
The sequence of heptagonal pyramidal numbers which also have the property of being square begins: :$0, 1, 196, \ldots$ {{expand|Further research needed}}
Conditions for Limit Function to be Limit Minimizing Function of Functional
Let $y$ be a real function. Let $J \sqbrk y$ be a functional. Let $\sequence {y_n}$ be a minimizing sequence of $J$. Let: :$\ds \lim_{n \mathop \to \infty} y_n = \hat y$ Suppose $J$ is lower semicontinuous at $y = \hat y$. Then: :$\ds J \sqbrk {\hat y} = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$
Smallest 10 Primes in Arithmetic Sequence
The smallest $10$ primes in arithmetic sequence are: :$199 + 210 n$ for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. These are also the smallest $8$ and $9$ primes in arithmetic sequence.
Sum of Cubes of 3 Consecutive Integers which is Square
The following sequence of $3$ consecutive positive integers have cubes that sum to a square: :$23, 24, 25$
Integer Greater than 205 is Sum of Distinct Primes of form 6 n + 1
Let $n$ be an integer greater than $205$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $6 n + 1$.
Numbers such that Tau divides Phi divides Sigma
The sequence of integers $n$ with the property that: :$\map \tau n \divides \map \phi n \divides \map \sigma n$ where: :$\divides$ denotes divisibility :$\tau$ denotes the $\tau$ (tau) function: the count of divisors of $n$ :$\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$ :$\sigma$ denotes the $\sigma$ (sigma) function: the sum of divisors of $n$ begins: :$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$ {{OEIS|A020493}}
Number of Representations as Sum of Two Primes
The number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\closedint {\dfrac n 2} {n - 2}$.
Integers whose Number of Representations as Sum of Two Primes is Maximum
$210$ is the largest integer which can be represented as the sum of two primes in the maximum number of ways. The full list of such numbers is as follows: :$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$ {{OEIS|A141340}} The list contains: :$n \le 8$ :$n \le 18$ where $2 \divides n$ :$n \le 48$ where $2 \times 3 \divides n$ :$n \le 90$ where $2 \times 3 \times 5 \divides n$ :$210 = 2 \times 3 \times 5 \times 7$ {{WIP|Add this sequence to the number pages}}
25 as Sum of 4 to 11 Squares
$25$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $4$ to $11$.
Cubes which are Sum of Three Cubes
The following cube numbers can be expressed as the sum of $3$ positive cube numbers: :$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$ {{OEIS|A066890}} with cube roots: :$6, 9, 12, 18, 19, 20, 24, 25, \ldots$ {{OEIS|A023042}} {{WIP|Add these sequences to the separate pages for the numbers themselves}}
Ritz Method implies Not Worse Approximation with Increased Number of Functions
Consider the Ritz method. Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$. Let $J \sqbrk {\eta_n} = \mu_n$. Then: :$\mu_n \ge \mu_{n + 1}$
Dissection of Cube into 3 Cubes using 8 Pieces
A cube can be dissected into $3$ smaller cubes by cutting it into $8$ pieces and reassembling them.
Smallest Multiplicative Magic Square is of Order 3
The order of the smallest multiplicative magic square is $3$, for example: {{:Multiplicative Magic Square/Examples/Order 3/Smallest}}
230 Fedorov Groups where Chiral Pairs are Distinct
There are $219$ Fedorov groups, if chiral copies are considered distinct.
Equivalence of Definitions of Amicable Pair
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Pair}}
Thabit's Rule
Let $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime. Then: :$\tuple {2^n a b, 2^n c}$ forms an amicable pair.
Smaller of Thabit Pair is Tetrahedral
Let $\tuple {m_1, m_2}$ be a Thabit pair such that $m_1 < m_2$. Then $m_1$ is a tetrahedral number.
Amicable Pair with Smallest Common Prime Factor 5
A amicable pair whose smallest common prime factor is greater than $3$ has the elements: :$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$ and: :$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 307 \, 071 \, 129$ This was at one point the smallest known counterexample to the observation that: :most amicable pairs consist of even integers :most of the rest, whose elements are odd, have both elements divisible by $3$.
Equivalence of Definitions of Amicable Triplet
Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Triplet}}
Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers
The largest positive integer which cannot be expressed as the sum of less than $37$ positive fifth powers is $223$: :$223 = 31 \times 1^5 + 6 \times 2^5$
Smallest Triple of Consecutive Sums of Squares
The smallest triple of consecutive positive integers each of which is the sum of two squares is: :$\tuple {232, 233, 234}$
No Quadruple of Consecutive Sums of Squares Exists
It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares.
Directed iff Filtered in Dual Ordered Set
Let $\struct {S, \preceq_1}$ be an ordered set. Let $\struct {S, \preceq_2}$ be a dual ordered set of $\struct {S, \preceq_1}$ Let $X \subseteq S$. Then: :$X$ is directed in $\struct {S, \preceq_1}$ : :$X$ is filtered in $\struct {S, \preceq_2}$.
Sum of Two Squares not Congruent to 3 modulo 4
Let $n \in \Z$ such that $n = a^2 + b^2$ where $a, b \in \Z$. Then $n$ is not congruent modulo $4$ to $3$.
Numbers not Expressible as Sum of Less than 9 Positive Cubes
The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes: {{begin-eqn}} {{eqn | l = 23       | r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3       | c =  }} {{eqn | l = 239       | r = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3       | c =  }} {{end-eqn}}
Largest Prime Factor of n squared plus 1
Let $n \in \Z$ be greater than $239$. Then the largest prime factor of $n^2 + 1$ is at least $17$.
Solution of Ljunggren Equation
The only solutions of the Ljunggren equation: :$x^2 + 1 = 2 y^4$ are: :$x = 1,  y = 1$ :$x = 239,  y = 13$ {{OEIS|A229384}}
Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239
Consider the indeterminate Diophantine equation: :$x^4 + y^4 = z^2 + 1$ When $x = 239$ and $x > y$, there are $3$ solutions: {{begin-eqn}} {{eqn | l = 239^4 + 104^4       | r = 58 \, 136^2 + 1       | c =  }} {{eqn | l = 239^4 + 143^4       | r = 60 \, 671^2 + 1       | c =  }} {{eqn | l = 239^4 + 208^4       | r = 71 \, 656^2 + 1       | c =  }} {{end-eqn}}
Number with over 240 Divisors is greater than 1,000,000
Let $n \in \Z$ be an integer with over $240$ divisors. Then: :$n > 1 \, 000 \, 000$
Sequence of 4 Consecutive Integers with Equal Number of Divisors
The following sequence of integers are sets of $4$ consecutive integers which all have the same number of divisors: :$\map \tau m = \map \tau {m + 1} = \map \tau {m + 2} = \map \tau {m + 3}$ where $\map \tau n$ denotes the $\tau$ function. :$242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \ldots$ {{OEIS|A039665}}
Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence
The  sequence of positive integers which are both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers begins: :$65, 250, \ldots$
Sum of 3 Cubes in 2 Distinct Ways
The sequence of positive integers which can be expressed as the sum of $3$ cubes numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 251       | r = 5^3 + 5^3 + 1^3       | rr= = 6^3 + 3^3 + 2^3 }} {{end-eqn}} {{OEIS|???}}
78,557 is Sierpiński
$78 \, 557$ is a Sierpiński number of the second kind.
Existence of Infinite Number of Numbers that are Riesel, Carmichael and Sierpiński
There exists an infinite number of integers which are simultaneously Sierpiński, Riesel and Carmichael.
Equivalence of Definitions of Balanced Prime
The following definitions of a balanced prime are equivalent:
Divisor of Fermat Number
Let $F_n$ be a Fermat number. Let $m$ be divisor of $F_n$.
Prime Decomposition of 6th Fermat Number
The prime decomposition of the $6$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^6} } + 1       | r = 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617       | c = Sequence of Fermat Numbers }} {{eqn | r = 274 \, 177 \times 67 \, 280 \, 421 \, 310 \, 721       | c =  }} {{eqn | r = \paren {3^2 \times 7 \times 17 \times 2^8 + 1} \times \paren {5 \times 47 \times 373 \times 2 \, 998 \, 279 \times 2^8 + 1}       | c =  }} {{end-eqn}}
Prime Decomposition of 7th Fermat Number
The prime decomposition of the $7$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^7} } + 1       | r = 340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457       | c = Sequence of Fermat Numbers }} {{eqn | r = 59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721       | c =  }} {{eqn | r = \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {3^5 \times 5 \times 12497 \times 733803 839347 \times 2^9 + 1}       | c =  }} {{end-eqn}}
Filtered iff Directed in Dual Ordered Set
Let $\left({S, \preceq_1}\right)$ be an ordered set. Let $\left({S, \preceq_2}\right)$ be a dual ordered set of $\left({S, \preceq_1}\right)$ Let $X \subseteq S$. Then $X$ is filtered in $\left({S, \preceq_1}\right)$  $X$ is directed in $\left({S, \preceq_2}\right)$
Filters equal Ideals in Dual Ordered Set
Let $L_1 = \left({S, \preceq_1}\right)$ be an ordered set. Let $L_2 = \left({S, \preceq_2}\right)$ be a dual ordered set of $L_1$. Then $\mathit{Filt}\left({L_1}\right) = \mathit{Ids}\left({L_2}\right)$ where :$\mathit{Filt}\left({L_1}\right)$ denotes the set of all filters of $L_1$, :$\mathit{Ids}\left({L_2}\right)$ denotes the set of all ideals of $L_2$.
Pépin's Test
Let $F_n = 2^{2^n} + 1$ be a Fermat number. Then $F_n$ is prime : :$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$
Triangular Fermat Number
The only one Fermat number which is triangular is $3$.
Fermat Number is not Square
There exist no Fermat numbers which are square.
Fermat Number is not Cube
There exist no Fermat numbers which are cubes.
Ideals equal Filters in Dual Ordered Set
Let $L_1 = \left({S, \preceq_1}\right)$ be an ordered set. Let $L_2 = \left({S, \preceq_2}\right)$ be a dual ordered set of $L_1$. Then $\mathit{Ids}\left({L_1}\right) = \mathit{Filt}\left({L_2}\right)$ where :$\mathit{Filt}\left({L_2}\right)$ denotes the set of all filters of $L_2$, :$\mathit{Ids}\left({L_1}\right)$ denotes the set of all ideals of $L_1$.
Power Set is Filter in Lattice of Power Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a inclusion lattice of power set of $X$. Then $\mathcal P\left({X}\right)$ is a filter on $L$.
Triplet in Arithmetic Sequence with equal Sigma
The smallest triple of integers in arithmetic sequence which have the same $\sigma$ (sigma) value is: :$\map \sigma {267} = \map \sigma {295} = \map \sigma {323} = 360$
Singleton of Set is Filter in Lattice of Power Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Then $\left\{ {X}\right\}$ is a filter on $L$.
Existence of Arbitrarily Long Aliquot Sequences
It is possible to construct an aliquot sequence of arbitrary length which is monotonically increasing.
Consecutive Triple of Repeated Digit-Products
The triplet of integers $281, 282, 283$ have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three, that is, $6$. There does not exist an set of four consecutive integers which also all end up at the same single digit.
Consecutive Powerful Numbers
The following pairs are of consecutive positive integers both of which are powerful: :$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$ {{OEIS|A060355}}
Smallest Multiple of 9 with all Digits Even
$288$ is the smallest integer multiple of $9$ all of whose digits are even.
Kaprekar's Symmetry
Let $n$ be a Kaprekar number with $D$ digits. Then $10^D - n$ is also a Kaprekar number.
Cyclic Permutation of Kaprekar Number
Let $n$ be a Kaprekar number of $k$ digits. Let $m$ be an integer formed from a cyclic permutation of the digits of $n$. Let $m$ be squared and the result split into $2$ parts, where the $2$nd part is of $k$ digits. Let these two parts be added, in the way of operating on a Kaprekar number. If the result is more than $k$ digits long, split that into $2$ parts, where the $2$nd part is of $k$ digits, and add the parts. The result will be another cyclic permutation of the digits of $n$.
Filters of Lattice of Power Set form Bounded Above Ordered Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes the set of all filters on $L$. Then $F$ is bounded above and $\top_F = \mathcal P\left({X}\right)$ where $\top_F$ denotes the greatest element of $F$.
Palindromic Primes in Base 10 and Base 2
The following $n \in \Z$ are prime numbers which are palindromic in both decimal and binary: :$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$ {{OEIS|A046472}} It is not known whether there are any more.
Ideals form Complete Lattice
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice. Let $\mathcal I = \struct {\mathit{Ids}\left({L}\right), \subseteq}$ be an inclusion ordered set, where $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$. Then $\mathcal I$ is complete lattice.
Filters form Complete Lattice
Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes set of all filters on $L$. Then $F$ is a complete lattice.
319 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$ or: :$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$
Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers
The following positive integers are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: === $79$ as Sum of $19$ Fourth Powers === {{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}} === $159$ as Sum of $19$ Fourth Powers === {{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $319$ as Sum of $19$ Fourth Powers === {{:319 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $399$ as Sum of $19$ Fourth Powers === {{:399 is not Expressible as Sum of Fewer than 19 Fourth Powers}}
Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331
Let $M \in \Z$ be an integer. Then there exists a positive integer $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no prime factors less than $331$.
Completely Irreducible implies Infimum differs from Element
Let $\struct {S, \preceq}$ be an ordered set. Let $p \in S$ such that :$p$ is completely irreducible. Then $\map \inf {p^\succeq \setminus \set p} \ne p$ where $p^\succeq$ donotes the upper closure of $p$.
Not Every Number is the Sum or Difference of Two Prime Powers
Not every positive integer can be expressed in the form $p^m \pm q^n$ where $p, q$ are prime and $m, n$ are positive integers.
Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element
Let $L = \struct {S, \preceq}$ be an ordered set. Let $p \in S$. Then $p$ is completely irreducible  :$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$ where $p^\succeq$ denotes the upper closure of $p$.
Poulet Number/Examples/341
The smallest Poulet number is $341$: :$2^{341} \equiv 2 \pmod {341}$ despite the fact that $341$ is not prime: :$341 = 11 \times 31$
Fourth Powers which are Sum of 4 Fourth Powers
The following positive integers are such that their fourth powers can be expressed as the sum of the fourth powers of $4$ other positive integers with no common factors: :$353, 651, 2487, 2501, 2829, \ldots$ {{OEIS|A039664}}
Sums of both 2 and 3 Consecutive Squares
The following are the smallest positive integers that are the sum of both $2$ and $3$ consecutive non-zero square numbers: :$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \, 925, \ldots$ {{OEIS|A007667}}