For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.
While studying a characteristic number of manifolds we noticed that the calculation was simply computing a multiple of a multinomial coefficient. We were, at the time, interested in computing the greatest common divisor of these characteristic numbers, and from that came this fun and interesting number theory problem. In this short paper, I will give the answer to this problem, which gives the greatest common divisor of certain multinomial coefficients.
For a function field $K$ and fixed polynomial $F\in K[x]$ and varying $f\in F$ (under certain restrictions) we give a lower bound for the degree of the greatest prime divisor of $F(f)$ in terms of the height of $f$, establishing a strong result for the function field analogue of a classical problem in number theory.
We establish consequences of the moving form of Schmidt's Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving $S$-unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin and others.
We introduce the weighted greatest common divisor of a tuple of integers and explore some of it basic properties. Furthermore, for a set of heights $\mathfrak w=(q_0, \ldots , q_n)$, we use the concept of the weighted greatest common divisor to define a height $\mathfrak{h} (\mathfrak p)$ on weighted projective spaces $\mathbb{WP}_{\mathfrak w}^n (k)$. We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem for heights on projective spaces.
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the prime factors of the argument, the explicit form of which is proven in this paper. Subsequently it is shown how the procedure can be generalized to the discrete Fourier transform of a function of the greatest common divisor. From this representation some interesting relations concerning the Euler totient function and generalizations thereof are established.
Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.
Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $τ$ and $μ$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.
The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan's sum, and on the other hand it can be written as a generalised convolution product of the identity with the totient function. We show that this arithmetic function of two integers (a,m) counts the number of elements in the set of ordered pairs (i,j) such that i*j is equivalent to a modulo m. Furthermore we generalise a dozen known identities for the toti…
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into probl…
Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. Leonetti and Sanna proved that $\mathcal{A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that \begin{equation*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{equation*} for all sufficiently large $x$.
We show that if two monic polynomials with integer coefficients have square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.
We prove inequalities involving greatest common divisors of functions at integral points with respect to numerically parallel divisors, generalizing a result of Wang and Yasufuku (after work of Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and the second author). After applying a result of Vojta on integral points on subvarieties of semiabelian varieties, we use geometry and the theory of heights to reduce to the (known) case of $\mathbb{G}_m^n$. In addition to proving results in a broader context than previously considered, we also study the exceptional set in this setting, for both the counting …
Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.
We apply Vojta's conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta's conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
Assume Vojta's Conjecture. Suppose $a, b, α,β\in \mathbb{Z}$, and $f(x),g(x) \in \mathbb{Z}[x]$ are polynomials of degree $d \ge 2$. Assume that the sequence $(f^{\circ n}(a), g^{\circ n}(b))_n$ is generic and $α,β$ are not exceptional for $f,g$ respectively, we prove that for each given $\varepsilon > 0$, there exists constant $C = C(\varepsilon,a,b,α,β,f,g)>0$, such that for all $n \ge 1$, we have $$\gcd(f^{\circ n}(a)-α, g^{\circ n}(b) -β) \le C\cdot\exp({\varepsilon\cdot d^n}). $$ We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as…
The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a rate of convergence to the normal distribution for the central limit theorem. Some generalizations are provided.
We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substitute…
Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\binom n1,\binom n2,\dots,\binom n{n-1}$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that the greatest common divisor of the binomial coefficients $\binom{2n}2,\binom{2n}4,\dots,\binom{2n}{2n-2}$ equals (a certain power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $0\le i\le j$. This note gives a concise proof of a tidy generalization of these facts.
Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of $\mathcal{P}_k$ exists, and we give a formula for $\text{r}(\mathcal{P}_k)$ in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given $k$ satisfies $\text{r}(\mathcal{P}_k) > 0$, and we provide upper and lower bounds for the counting function of the set of such $k$'s. As an application of our results, we give …
For a fixed $b\in \mathbb{N}=\{1,2,3,\dots\}$, Goins et al. \cite{Harris} defined the concept of $b$-visibility for a lattice point $(r,s)$ in $L=\mathbb{N}\times \mathbb{N}$ which states that $(r,s)$ is $b$-visible from the origin if it lies on the graph of $f(x)=ax^b$, for some positive $a\in \mathbb{Q}$, and no other lattice point in $L$ lies on this graph between $(0,0)$ and $(r,s)$. Furthermore, to study the density of $b$-visible points in $L$ Goins et al. defined a generalization of greatest common divisor, denoted by $\gcd_b$, and proved that the proportion of $b$-visible lattice point…
Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant compositionally independent polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $λ$ with the property that there is an $n$ such that $(x - λ)$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$.
Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$ as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We consider the generalization of this matrix where the elements are in the form $f\big(i,(i,j)\big)$.
Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of …
Indraprastha Institute of Information Technology（印度普拉斯坦萨信息技术学院）；Chennai Mathematical Institute（马德拉斯数学研究所）；Institut Fourier, Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学傅里叶研究所）
A Greatest Common Divisor Criterion of Certain Binomial Coefficients
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences（数学科学学院，中国科学院大学；中国科学院数学与系统科学研究院）
We improve some results on the size of the greatest prime factor of integers of the form ab+1, where a and b belong to finite sets of integers with rather large density.
We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs for some fixed values of $k$. In addition, we exhibit the run of 11 consecutive positive integers each with exactly 36 divisors, the run of 13 consecutive positive integers each with exactly 12 divisors, the run of 17 consecutive positive integers each with exactly 24 divisors, and the run of 17 consecutive positive integers each with exactly 48 divisors.
For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $σ_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function $σ_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q.
A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.
We study the function $Θ(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results.
This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.
Let H(x,y,z) be the number of integers $\le x$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $\le x$ with exactly one such divisor. When y and z are close, it is expected that H_1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H_1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223.
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
A positive integer $n$ is called $φ$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $φ$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.
Suppose that $θ$ is irrational. Then almost all elements $ν\in {\mathbb Z}[θ]$ that may be written as a $k$-fold product of the shifted integers $n+θ$ $(n\in {\mathbb N})$ are thus represented essentially uniquely.
Every positive integer greater than a positive integer $r$ can be written as an integer that is the sum of powers of $r$. Here we use this to prove the conjecture posed by Ronald Graham, B. Rothschild and Joel Spencer back in the nineteen nineties, that the van der Waerden number with $r$ colorings and with arithmetic progressions of $k$ terms, has a certain upper bound. Our proof does not need the application of double induction, constructive methods of proof or combinatorics, as applied to sets of integers that contain some van der Waerden number as an element. The proof instead derives from…
Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1μ_1+a_2μ_2+\ldots+a_sμ_m$, where positive integers $μ_1,μ_2,\ldots,μ_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $μ_1,μ_2,\ldots,μ_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be writt…
We improve existing estimates of moments of the Riemann zeta function. As a consequence, we are able to derive new estimates for the asymptotic behaviour of $\sum_{N α\le x} \mathfrak{t}_k(α)$, where $N$ stands for the norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor function on Gaussian integers.
Let $\taue_k \colon \Z\to\Z$ be a multiplicative function such that $ \taue_k(p^a) = \sum_{d_1... d_k=a} 1 $. In the present paper we introduce generalizations of $\taue_k$ over the ring of Gaussian integers $\Zi$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.
Motivated by a question of Venkataramana, we consider the greatest common divisor of $ϕ(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we establish that this gcd may be bounded just in terms of the degree of the polynomial $f$. Unconditionally we establish such a bound for quadratic polynomials, as well as polynomials that split completely into linear factors. The paper also addresses a question of Calegari, and establishes that there are infinitely many $n$ such that $n^2+1$ is not divisible by any…
We have found maximum possible runs of consecutive positive integers each having exactly $k$ divisors for some fixed values of $k$. In addition, we exhibit the run of 10 consecutive positive integers each having exactly 12 divisors and two runs of 14 consecutive positive integers each having exactly 24 divisors.
We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1 and b-1 are coprime, then a^k-1 and b^k-1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do this we use a result of Lang's on the finiteness of torsion points on algebraic curves. We also give a matrix analogue of these results, where for a unimodular matrix A, we look at the greatest common divisor of th…
