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https://mathoverflow.net/questions/510656/can-gcdnk-pm-1-hspace2mm-n-pm-11-have-arbitrarily-many-but-finitel nt.number theory - Can $\gcd(n^k \pm 1, \hspace{2mm} n! \pm 1)1$ have arbitrarily many but... Fix an integer $k \geq 2$ and let $n \geq 2$ be integer as well. Let $\lambda_1,\lambda_2 \in \{-1, 1 \}$, and then consider the set $$... nt number theory k pm gcd 1 2mm https://mathoverflow.net/questions/338061/bounds-for-the-number-of-prime-numbers-less-than-the-eulers-factor-the-radical nt.number theory - Bounds for the number of prime numbers less than the Euler's factor, the radical... As tell us the Wikipedia section dedicated to Odd perfect numbers (please, see also the related references if you need it), any perfect number has the form... nt number theory prime numbers bounds less euler https://mathoverflow.net/questions/510568/josef-plemeljs-work-on-the-quintic-case-of-flt nt.number theory - Josef Plemelj's work on the quintic case of FLT - MathOverflow It is well known that Dirichlet proved that the equation $x^5+y^5=z^5$ has no non-trivial integer points, by using properties of the norm-euclidean field... nt number theory josef work quintic case https://mathoverflow.net/questions/509928/equivalence-of-pair-of-integer-coefficients nt.number theory - Equivalence of pair of integer coefficients - MathOverflow Please note that this question was completely reworked. Let $S(n,k)$ be an integer coefficients such that $$ S(n,k) = \sum\limits_{j=0}^{k} \frac{n!}{j!} {n-j... nt number theory equivalence pair integer coefficients https://mathoverflow.net/questions/111196/name-of-a-conjecture-on-difference-of-prime-numbers nt.number theory - Name of a conjecture on difference of prime numbers? - MathOverflow Hello Dear there is a conjecture for which I do not know how it is called. The conjecture is: Every even number can be always written as the difference between... nt number theory prime numbers name conjecture difference https://mathoverflow.net/questions/471098/1-1-map-on-the-0-1-k nt.number theory - 1-1 map on the $\{0,1\}^k$ - MathOverflow nt number theory 1 map 0 k mathoverflow https://mathoverflow.net/questions/377964/convergence-of-the-series-involving-mobius-functions-sum-k-d-mud-x-kd nt.number theory - Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$... (I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more... nt number theory mu x convergence series involving https://mathoverflow.net/questions/505922/fast-and-simple-recurrence-for-sums-over-powers-and-factorials nt.number theory - Fast and simple recurrence for sums over powers and factorials - MathOverflow Let $f(n,m)$ be a function such that $$ f(n,m) = mf(n-1,m) + 1, \\ f(0,m) = 1. $$ $T(n,m)$ be a coefficients such that $$ T(n,m) = n! \sum\limits_{k=1}^{n}... nt number theory fast simple recurrence sums https://mathoverflow.net/search?q=user:3635+[nt.number-theory] Posts matching 'user:3635 [nt.number-theory]' - MathOverflow posts matching user nt number theory 3635 mathoverflow https://mathoverflow.net/questions/509384/is-the-limit-of-this-hybrid-sum-integral-sequence-always-transcendental nt.number theory - Is the limit of this hybrid sum-integral sequence always transcendental? -... Let $f: \mathbb{R}^+ \to \mathbb{R}$ be a $C^\infty$ function where all moments $M_n = \int_0^\infty x^n f(x) dx$ are rational. Consider the limit: $$L =... nt number theory limit hybrid sum integral https://mathoverflow.net/questions/510394/identity-with-stirling-numbers-of-the-first-kind nt.number theory - Identity with Stirling numbers of the first kind - MathOverflow Let ${n \brack k}$ be the unsigned Stirling numbers of the first kind. With the luck of intuition and after a lot of numerical experiments I conjecture that $$... nt number theory stirling numbers identity first kind https://mathoverflow.net/questions/469606/on-a-a089039-and-pair-of-sequences-with-simple-recursion nt.number theory - On a A089039 and pair of sequences with simple recursion - MathOverflow Let $a(n)$ be A089039 (i.e., number of circular permutations of $2n$ letters that are free of jealousy). Here $$ a(n) =... nt number theory pair sequences simple recursion https://mathoverflow.net/questions/510469/closed-form-for-certain-integer-coefficients-based-on-a075856-using-a291980 nt.number theory - Closed form for certain integer coefficients based on A075856 using A291980 -... Let $a(n)$ be A074059, i.e., an integer sequence known as dimension of the cohomology ring of the moduli space of $n$-pointed curves of genus $0$ satisfying... nt number theory closed form certain integer coefficients https://mathoverflow.net/questions/tagged/nt.number-theory?tab=Unanswered Unanswered 'nt.number-theory' Questions - MathOverflow nt number theory questions mathoverflow unanswered https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat nt.number theory - Zagier's one-sentence proof of a theorem of Fermat - MathOverflow Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares. The proof defines an... nt number theory one sentence proof theorem fermat https://mathoverflow.net/questions/510474/covering-cyclic-groups-with-translates-of-a-large-subset nt.number theory - Covering cyclic groups with translates of a large subset - MathOverflow I am inteterested in the following very basic question. Given a subset $A\subset \mathbb Z_{n}$ with $|A|=0.9n$, is it possible to find a translate $x+A$ so... nt number theory covering cyclic groups translates https://mathoverflow.net/questions/509735/was-fermats-last-theorem-known-for-infinitely-many-primes-before-wiles nt.number theory - Was Fermat's Last Theorem known for infinitely many primes before Wiles? -... Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes... nt number theory fermat last theorem known https://mathoverflow.net/questions/510610/on-x32x2-phi2-0-the-snub-dodecahedron-and-the-octic-with-n-real-roots nt.number theory - On $x^3+2x^2-\phi^2=0$, the snub dodecahedron, and the octic with $n$ real roots... (Update: Gave example of a suitable octic in the last section.) This is a tangent from a previous polyhedra post. I came across the solvable sextic, $$x^6 +... nt number theory x 3 2x 2 phi 0 https://mathoverflow.net/questions/tagged/nt.number-theory Newest 'nt.number-theory' Questions - MathOverflow nt number theory questions mathoverflow newest https://mathoverflow.net/questions/509806/closed-form-for-a135494 nt.number theory - Closed form for A135494 - MathOverflow Let $f(n)$ be an integer function such that $$ f(n) = -1, \\ f(1) = 1. $$ $T(n,k)$ be A135494, i.e, an integer coefficients known as Bell transform of $f(n)$.... nt number theory closed form a135494 mathoverflow https://mathoverflow.net/questions/510374/recurrence-for-certain-family-of-coefficients nt.number theory - Recurrence for certain family of coefficients - MathOverflow Let $a(n)$ be an arbitrary sequence with exponential generating function $A(x)$ and $a(1) = 1$. $T(n,k)$ be a coefficients (represented as a triangle) with... nt number theory recurrence certain family coefficients https://mathoverflow.net/questions/18817/does-2m-3n-r-have-finitely-many-solutions-for-every-r nt.number theory - Does $2^m = 3^n + r$ have finitely many solutions for every $r$? - MathOverflow Is it true that for every integer $r$, the equation $2^m = 3^n + r$ has at most a finite number of integer solutions? I understand that this is a special case... nt number theory 2 3 many solutions https://mathoverflow.net/questions/132648/the-erd%c5%91s-tur%c3%a1n-conjecture-or-the-erd%c5%91s-conjecture nt.number theory - The Erdős–Turán conjecture or the Erdős conjecture? - MathOverflow This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:... nt number theory conjecture mathoverflow https://mathoverflow.net/questions/152665/is-there-a-lower-bound-for-the-first-non-trivial-sequence-of-consecutive-integer nt.number theory - Is there a lower bound for the first non-trivial sequence of consecutive... Using the Chinese Remainder Theorem, it is very straight forward to find a sequence of consecutive integers starting at $x$ where each of the first $n$ prime... nt number theory lower bound first non trivial sequence https://mathoverflow.net/questions/509633/is-the-xin-prod-k-1-infty-textlcm1-2-dots-lfloor-n1-k-r nt.number theory - Is the $\Xi(n) = \prod_{k=1}^{\infty} \text{lcm}(1, 2, \dots, \lfloor n^{1/k}... In this previous discussion, it was demonstrated that the standard Least Common Multiple sequence $\text{lcm}(1, 2, \dots, n)$ is not a subset of the highly... nt number theory k 1 xi prod https://mathoverflow.net/questions/202316/small-quotients-of-smooth-numbers nt.number theory - Small quotients of smooth numbers - MathOverflow Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in... nt number theory smooth numbers mathoverflow small https://mathoverflow.net/questions/507990/do-there-exist-sequences-with-liminf-x-n1-x-n-3-for-which-no-alpha-v nt.number theory - Do there exist sequences with $\liminf x_{n+1}/x_{n}=3$ for which no $\alpha,... Background Let $(x_n)_{n=1}^{\infty}$ be a sequence of positive integers and let $$\liminf_{n \to \infty}\frac{x_{n+1}}{x_n}=\lambda.$$ Consider the following... nt number theory 1 3 exist sequences alpha https://mathoverflow.net/questions/509867/identities-with-stirling-numbers-of-both-kinds nt.number theory - Identities with Stirling numbers of both kinds - MathOverflow With the luck of intuition, I conjecture that $$ {n+m+1 \brack m+1} = (-1)^n \sum\limits_{k=0}^{n} \left[ \left[ \sum\limits_{i=0}^{k} (-1)^{k+i} 2^{k-i}... nt number theory stirling numbers identities kinds mathoverflow https://mathoverflow.net/questions/508850/quadratic-polynomials-and-riemanns-zeta-function nt.number theory - Quadratic polynomials and Riemann's zeta function - MathOverflow Motivated by my recent postings, here I propose some conjectural series for Riemann's zeta function with summands involving binomial coefficients and quadratic... nt number theory zeta function quadratic polynomials riemann https://mathoverflow.net/questions/509218/is-this-pratt-tree-decomposition-known-in-literature-maybe-under-a-different-na nt.number theory - Is this Pratt-tree decomposition known in literature, maybe under a different... Let $m_p(n)$ be the number of times, counted with multiplicity, that the prime $p$ occurs in all Pratt trees of $n$. A Pratt tree for a prime $p$ is known. For... nt number theory pratt tree decomposition known https://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q nt.number theory - Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$? -... Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection? nt number theory polynomial bijection mathbb q https://mathoverflow.net/questions/509729/identity-using-varphin-dn-and-sigman nt.number theory - Identity using $\varphi(n), d(n)$ and $\sigma(n)$ - MathOverflow Let $\varphi(n)$ be the Euler totient function. $d(n)$ be the number of divisors of $n$. $\sigma(n)$ be the sum of the divisors of $n$. $a(n)$ be A344598,... nt number theory identity using sigma mathoverflow https://mathoverflow.net/questions/474520/recursion-for-the-sum-with-stirling-numbers-of-both-kinds nt.number theory - Recursion for the sum with Stirling numbers of both kinds - MathOverflow Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $n \brace k$ be a Stirling number of the second kind. Let $$ f(n,m,i) =... nt number theory stirling numbers recursion sum kinds https://mathoverflow.net/questions/510629/asymptotic-behaviour-of-difference-set-of-smooth-numbers nt.number theory - Asymptotic behaviour of difference set of smooth numbers - MathOverflow As usual, a $k$ smooth number is an integer that has no prime factor exceeding $k$. For each fixed constant $k$, let $A$ be the set of $k$ smooth numbers. The... nt number theory difference set smooth asymptotic behaviour numbers mathoverflow https://mathoverflow.net/questions/117390/polynomial-bijection-from-zxz-to-z nt.number theory - Polynomial bijection from ZxZ to Z? - MathOverflow It is known that the polynomial $f(n,m)=\frac{1}{2}(n+m)(n+m+1)+m$ defines bijection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ (Put pairs of $\mathbb{N}$ into... nt number theory polynomial bijection z mathoverflow