On exponentially coprime integers

On Sequences Containing at Most 4 Pairwise Coprime Integers

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers…

On a conjecture of Erdős about sets without $k$ pairwise coprime integers

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Z…

Inhomogeneous approximation with coprime integers and lattice orbits

We obtain variants of the classical Minkowski Theorem on inhomogeneous approximation where we require moreover that the solutions $p, q$ be coprime integers. We link the subject with density exponents of lattice orbits in the real plane.

On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers

We use the group $(\Z^2,+)$ and two associated homomorphisms, $τ_0, τ_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the parents of the same integer pair in $T$. Our main result is a proof that for $x_i \in \{0,1\}$, the sum of the pair $τ_{x_1}τ_{x_2}... τ_{x_n} [1,2]$ is equal to the sum of the pair $τ_{x_n}τ_{x_{n-1}} ... τ_{…

Distribution of the k-regular partition function modulo composite integers M

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$.

Coprime matchings

We prove that there is a matching between 2 intervals of positive integers of the same even length, with corresponding pairs coprime, provided the intervals are in $[n]$ and their lengths are $>c(\log n)^2$, for a positive constant $c$. This improves on a recent result of Bohman and Peng. As in their paper, the result has an application to the lonely runner conjecture.

多项式定义的可乘函数在互素剩余类中的分布

Distribution in coprime residue classes of polynomially-defined multiplicative functions

本文研究多项式定义的可乘函数在互素剩余类中的分布，建立了模数大范围一致条件下的等分布结果，并证明欧拉函数值在特定模数下渐近等分布。

Complete Subgraphs of the Coprime Hypergraph of Integers III: Construction

The coprime hypergraph of integers on $n$ vertices $CHI_k(n)$ is defined via vertex set $\{1,2,\dots,n\}$ and hyperedge set $\{\{v_1,v_2,\dots,v_{k+1}\}\subseteq\{1,2,\dots,n\}:\gcd(v_1,v_2,\dots,v_{k+1})=1\}$. In this article we present ideas on how to construct maximal subgraphs in $CHI_k(n)$. This continues the author's earlier work, which dealt with bounds on the size and structural properties of these subgraphs. We succeed in the cases $k\in\{1,2,3\}$ and give promising ideas for $k\geq 4$.

On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $\mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd(F(n), G(n)) = h$. We prove that $\mathcal{A}_{F,G,h}$ has a natural density. Moreover, assuming $F$ is non-degenerate and $G$ has no fixed divisors, we show that $\mathbf{d}(\mathcal{A}_{F,G,1}) = 0$ if and only if $\mathcal{A}_{F,G,1}$ is finite.

On factorizations into coprime parts

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^β$ and $G(n)^β$ up to $x$ for all real $β$ and the asymptotic bounds for $f(n)^β$ and $g(n)^β$ for all negative $β$.

On a problem of Arnold: the average multiplicative order of a given integer

For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l_g(p) as p <= x ranges over primes.

Coprime-Universal 二次型

Coprime-Universal Quadratic Forms

给定素数 $p>3$，本文刻画了对 $p$ 是 coprime-universal 的正定整二次型，推广了 $290$ 定理，并在 GRH 下证明了特定情况，最后探讨了等差数列的表示问题。

关于分量两两不互素的正整数 $k$ 元组的渐近密度

On the asymptotic density of $k$-tuples of positive integers with pairwise non-coprime components

本文利用多变量算术函数卷积方法，研究分量两两不互素及含指定数量互素对的正整数 $k$ 元组计数，给出渐近公式并回答、推广前人结果。

On certain arithmetic functions involving exponential divisors

The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. In the paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors and exponentially coprime integers.

Coprime Mappings and Lonely Runners

For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number $t$ such that $ 1/(n+1) \le \{ v_i t\} \le n/(n+1) $ for $ i = 1, \dots, n$. In this paper we prove that if $ ε>0 $ and $n$ is sufficiently large (relative to $ε$) then such a $t$ exists for any collection of positive integers $ v_1 < v_2 < \dots < v_n$ such that $ v_n < (2-ε)n$. This is an approximate version of a natural next step for the study of the lone…

关于非CM椭圆曲线的共素减少密度

On the Density of Coprime Reductions of Elliptic Curves

研究非CM椭圆曲线在良好约简素数p下，其阶数互质的自然密度，提出猜想并证明相关级数收敛及近似欧拉乘积展开。

互素数的 Bézout 系数近似二次 Bézier 曲线

Bézout coefficients of coprime numbers approximate quadratic Bézier curves

本文证明，对于非负整数点 $(p,q)$（$p\ eq q$），由三点定义的二次 Bézier 曲线近似于由邻域内互素数 Bézout 系数构成的线段族的包络。

Congruences for sums of powers of an integer

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where $m(q,e)$ is "large". We also pay special attention to the situation where $e$ is a prime power.

Pairwise non-coprimality of triples

We say that $(a_1,...,a_k)$ is pairwise non-coprime if $\gcd(a_i,a_j) \ne 1$ for all $1 \le i <j \le k$. Let $a_1,a_2,a_3$ be positive integers less than $H$. We obtain an asymptotic formula for the number of $(a_1,a_2,a_3)$ that are pairwise non-coprime. The probability that a randomly chosen unbounded positive integer triple is pairwise non-coprime is approximately 17.4%. Let $φ(n)$ be the Euler totient function. We also give an upper bound on the error term in an asymptotic formula for $\sum_{n=1}^H (φ(n)/n)^m$ for $m \ge 2$ and as $H \rightarrow \infty$.

与6互素的二项式系数

Binomial coefficients coprime to 6

本文针对d含至少两个不同素因子时无有意义界的问题，首次给出与6互素的二项式系数数量的非平凡界，推进了例外集合规模的研究。

On the discriminant of pure number fields

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The paper contains an explicit formula for the discriminant of $K$ involving only the prime powers dividing $a,n$.

On a conjecture of Pomerance

We say that k is a P-integer if the first phi(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P-integers and conjectured that 30 is the largest P-integer. We prove the conjecture assuming the Riemann Hypothesis. We further prove that there is no P-integer between 30 and 10^11 and none above 10^3500.

Character sums over shifted primes

For integer $q$, let $χ$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}Λ(n)χ(n+a),$$ where $Λ(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski

Counting carefree couples

A pair of natural numbers (a,b) such that a is both squarefree and coprime to b is called a carefree couple. A result conjectured by Manfred Schroeder (in his book `Number theory in science and communication') on carefree couples and a variant of it are established using standard arguments from elementary analytic number theory. Also a related conjecture of Schroeder on triples of integers that are pairwise coprime is proved.

A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.

Compositions with 3 Pairwise Coprime Parts

How many ways can we write $n$ as a sum of $3$ positive integers, no pair of which share a common factor? We express this quantity in terms of the number of solutions to a certain class of linear Diophantine equations. This allows us to show that there are $$ \prod_{p \mid n} \left( 1- \frac{1}{p^2} \right) \prod_{q \nmid n} \left( 1- \frac{3}{q^2} \right) \frac{n^2}{2} + O(n^{3/2+o(1)}) $$ such compositions, where the products are over primes that respectively do and don't divide $n$. This strengthens the previous result of Bubbolini, Luca, and Spiga (arXiv:1202.1670)

素数与多边形数

Primes and polygonal numbers

本文证明了多边形数线性组合产生无穷多素数且其倒数和收敛的条件，并指出形如 $am^2+bn^2$ 的素数中存在任意长度的等差数列。

A Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at most three positive integer solutions $(x,y,z)$.

An Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations II

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$.

Toda素数

Toda primes

该研究提出Toda素数的定义，猜想每个正整数至少有两个Toda素数，给出至少存在一个的部分证明，并探讨其与Bernoulli数分母的关联及对Sophie Germain素数的推广。

The first elements of the quotient of a numerical semigroup by a positive integer

Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved version of Johnson's semi-explicit formula for the Frobenius number $g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give an explicit formula for a particular class of numerical semigroups.

On primes in arithmetic progression having a prescribed primitive root. II

Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.

关于完美幂与整数S-单位的差异

On the difference between perfect powers and integral $S$-units

研究通过建立有效下界，探讨了正整数z与整数S-单位之间的差异，揭示了其最大素因子随z^d增长的趋势。

The order of the reductions of an algebraic integer

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K(ζ_m, \sqrt[n]{a}) with n\leq m, which are obtained by adjoining a root of unity of order l^m and the l^n-th roots of a, as this is needed for computing the above density.

On the number of solutions of the generalized Ramanujan-Nagell equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$}

Let $D_{1}$, $D_{2}$ be coprime odd integers with min$(D_{1},D_{2})>1$, and let $N(D_{1},D_{2})$ denote the number of positive integer solutions $(x, m, n)$ of the equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$. In this paper, we prove that $N(D_{1},D_{2})\leq 2$ except for $N(3,5)=N(5,3)=4$ and $N(13,3)=N(31,97)=3$.

Congruence ABC implies ABC

The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c) with 16 | abc, then the full ABC conjecture holds. We extend that result to show that, for every integer N, the "congruence ABC conjecture" that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.

Deterministic factorization of sums and differences of powers

Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime factorization of $N$ in \[ \mathcal{O}(\text{M}_{\text{int}}(N^{1/4}\sqrt{\log N})), \] $\text{M}_{\text{int}}(k)$ denoting the cost for multiplying two $k$-bit integers. This result is better than the currently best known general bound for the runtime complexity for deterministic integer factorization.

On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n

In this paper, we give all the solutions of the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y coprime, except for the case when alpha.x is odd and beta is even.

On the distribution of the order over residue classes

The multplicative order of an integer g modulo a prime p, with p coprime to g, is defined to be the smallest positive integer k such that g^k is congruent to 1 modulo p. For fixed integers g and d the distribution of this order over residue classes mod d is considered as p runs over the primes. An overview is given of the most significant of my results on this problem obtained (mainly) in the three part series of papers `On the distribution of the order and index of g (modulo p) over residue classes' I-III (appeared in the Journal of Number Theory, also available from the ArXiv).

Reciprocity Relations for Summations of Squares of Floor Functions and Fractional Parts of Fractions

Given positive coprime integers $a$ and $b$ and a natural number $h$, we obtain reciprocity relations which can be used to quickly evaluate summations like $\sum_{i=1}^{h} \{\frac{ib}{a}\}^2$ and $\sum_{i=1}^{h} \lfloor \frac{ib}{a} \rfloor^2$, where $\lfloor x \rfloor$ and $\{x\}$ denote the floor function and the fractional part of $x$, respectively.

On the density of primes in arithmetic progression having a prescribed primitive root

Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this note we explicitly evaluate this density and give some applications of this result.

On a modular form of Zaremba's conjecture

We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a/q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q = O(p^{C})$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a/q$ are bounded by two.

On the Diophantine equation x^2+2^a.3^b.11^c=y^n

In this note, we find all the solutions of the Diophantine equation x^2 +2^a.3^b.11^c=y^n in nonnegative integers a, b, c, x, y, n>= 3 with x and y coprime.

Joint distribution in residue classes of polynomial-like multiplicative functions

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as complementary to work of Narkiewicz, who obtained comprehensive results for fixed moduli.

On (a,b) Pairs in Random Fibonacci Sequences

We study the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if x is the parent of y, then y has two children, namely |x-y| and x+y. This tree was studied by Benoit Rittaud who proved that any pair of integers a,b that are coprime occur as a parent-child pair infinitely often. We extend his results by determining the probability that a random infinite walk in this tree contains exactly one pair (1,1), that being at the r…

A Lucas-Lehmer approach to generalised Lebesgue-Ramanujan-Nagell equations

We describe a computationally efficient approach to resolving equations of the form $C_1x^2 + C_2 = y^n$ in coprime integers, for fixed values of $C_1$, $C_2$ subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

指数为一的阻塞器与Mordell-Weil构造的欧拉砖

Exponent-one blockers and a Mordell-Weil construction of Euler bricks

本文研究欧拉砖的构造，提出指数为一的阻塞现象，并利用椭圆纤维化方法生成Mordell-Weil组合，证明所有生成的欧拉砖均不满足完美欧拉砖条件。

Primes in a prescribed arithmetic progression dividing the sequence a^k+b^k

Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.

带数字限制的Waring问题

Waring's problem with restricted digits

本文研究了带数字限制的Waring问题，证明了在给定进制下仅使用两个互质数字时，足够大的整数均可表示为至多 $b^{160 k^2}$ 个特定集合中元素的 $k$ 次幂之和。

On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work, we calculate $r_2(K)$, the $2$-rank of the class group of $K$, in terms of the number of prime divisors of $d$ that decompose or remain inert in $\mathbb{Q}(\sqrt{p})$, then we will deduce forms of $d$ satisfying $r_2(K)=2$. In the last case, the $4$-rank of the class group of $K$ is given too.

A characterization of the Frobenius problem and its application to arithmetic progressions

In the Frobenius problem we are given a set of coprime, positive integers $a_1, a_2,...,a_k$, and are interested in the set of positive numbers NR that have no representation by the linear form $\sum_i a_ix_i$ in nonnegative integers $x_1, x_2,...,x_k$. We give a functional relationship that completely characterizes the set NR, and apply it to the case when the numbers are in an arithmetic progression.

Sum of many dilates

We show that for any coprime integers $λ_1 , \ldots , λ_k$ and any finite $A \subset \mathbb{Z}$, one has $$|λ_1 \cdot A + \ldots + λ_k \cdot A| \geq (|λ_1| + \ldots + |λ_k|)|A|- C,$$ where $C$ only depends on $λ_1 , \ldots , λ_k$.

Progression-free sets in Z_4^n are exponentially small

We show that for integer $n>0$, any subset $A \subset Z_4^n$ free of three-term arithmetic progressions has size $|A| < 4^{c n}$, with an absolute constant $c \approx 0.926$.

$abc$猜想几乎总是成立的

The $abc$ conjecture is true almost always

本文介绍de Bruijn的经典估计，证明几乎所有互素整数三元组满足$abc$猜想（至多$O(N^{2/3})$个反例），并为Browning等人给出的$O(N^{33/50})$改进估计（1962年来首次幂次节省）提供背景。