Kevin Ford, Mikhail R. Gabdullin
We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$
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Using Watson's terminating $_8ϕ_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.
Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$, and let $\mathbb{Z}_n^{\ast}$ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of $c\in \mathbb{Z}_n$ as the sum of $k$ units. Recently, Yang and Tang in [Q. Yang, M. Tang, On the addition of squares of units and nonunits modulo $n$, J. Number Theory., 155 (2015) 1--12] gave a formula for the number of solutions of the equation $x_1^2+x_2^2=c$ with $x_{1},x_{2}\in \mathbb{Z}_n^{\ast}$. In this paper, we generalize this result. We find an explicit formula for the number of solutions…
In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.
We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.
A note on the order of the Tate--Shafarevich group modulo squares
本文研究了阿贝尔簇的Tate--Shafarevich群模有理平方的阶，证明了每个无平方因子的自然数均可作为某个阿贝尔簇Tate--Shafarevich群的非无平方因子部分出现，从而验证了W. Stein的猜想。
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
Formulas For The Square Root Modulo p
该研究针对形如p=2^k n+1（n为奇数）的素数，提出构造模p平方根函数在有限域F_p上多项式表示的方法，并给出k为2、3、4时的具体公式。
The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtwängler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and $S$-singular elements (modulo squares) in a global function field.
We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum\limits_{n=0}^{p-1}\dbinom{2n}{n} \equivη_{p}\mod p^{2},$$ $$\sum\limits_{n=0}^{rp-1}\dbinom{2n}{n} \equivη_{p}\sum\limits_{n=0}^{r-1}\dbinom {2n}{n}\mod p^{2}$$ and $$\sum\limits_{n=0}^{rp-1}\sum\limits_{m=0}^{sp-1}\dbinom{n+m}{m}^{2} \equivη_{p} \sum\limits_{m=0}^{r-1}\sum\limits_{n=0}^{s-1}\dbinom{n+m}{m}^2\mod p^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $η_{p}= \begin{cases} 0, &\text{if }p\equiv0\mod 3;\\ 1, & \text{if }p\equiv1\mod 3;\\ -1, &\text{if }p\equiv2…
Factorial residues modulo a prime: beyond the square-root bound
本研究针对素数模阶乘剩余集的大小下界问题，利用有限域阶乘恒等式、分式线性映射交点性质与Stevens-de Zeeuw点线相交定理，将下界从平方根级改进至\\(p^{8/15}\\)量级。
Square-free orders for CM elliptic curves modulo $p$ in short intervals
本文研究有理数域上CM椭圆曲线模素数的无平方因子阶计数问题，改进了已有渐近公式，得到无条件短区间渐近结果，且适用区间比广义黎曼假设下的更短，同时优化了CM椭圆曲线循环性问题的短区间变体结论。
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets
本文探讨了Gallagher筛法在模素数平方下的应用局限，并提出通过转换余数信息以提升筛法效果，同时研究了Hilbert立方体在不同乘法定义集中的最大维度，得出了相关上界。
We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.
Although squaring integers is deterministic, squares modulo a prime, $p$, appear to be random. First, because they are all generated by the multiplicative linear congruential equation, $x_{i+1} = g^2 x_i \mod p$, where $x_0 = 1$ and $g$ is any primitive root of $p$, a pseudorandom number heuristic suggests that they are, in fact, unpredictable. Moreover, one type of cryptography makes use of discrete algorithms, which depends on the difficulty of solving $a = g^n$ for $n$ given $a$ and $g$. This suggests that the squares, which are exactly the even powers of $g$, are hard to identify. On the o…
It is widely believed that the parity of the partition function $p(n)$ is ``random.'' Contrary to this expectation, in this note we prove the existence of infinitely many congruence relations modulo 4 among its values. For each square-free integer $1<D\equiv 23\pmod{24},$ we construct a weight 2 meromorphic modular form that is congruent modulo 4 to a certain twisted generating function for the numbers $p\big(\frac{Dm^2+1}{24}\big)\pmod 4$. We prove the existence of infinitely many linear dependence congruences modulo 4 among suitable sets of holomorphic normalizations of these series. These r…
$ $The aim of this thesis is to lower the bound on square-free primitive roots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive root modulo $p$. We have proven the following two theorems. Theorem 0.1. $$g^{\Box}(p) < p^{0.88}\quad\text{for all primes }p.$$ Theorem 0.2. $$g^{\Box}(p) < p^{0.63093}\quad\text{for all primes } p < 2.5\times10^{15} \text{ and }p > 9.63\times10^{65}.$$ Theorem 0.1 shows an improvement in the best known bound while Theorem 0.2 shows for which primes we can prove the theoretical lower bound. After some introductory information in Chapter 1, we will …
On the Birch-Swinnerton-Dyer quotients modulo squares
设A是数域K上的阿贝尔簇. K的扩张K_i的L-函数L（A/K_i，s）之间的恒等式导出了Birch-Swinnerton-Dyer恒等式之间的关系。我们证明了这些关系的模有限性的沙，并给出了类似的声明Selmer群。在此基础上，我们发展了一种方法，用于确定A在K的扩张上的秩的各种组合的奇偶性。作为应用之一，我们在假设Sha[6^ \infty ]有限性和对素数大于2和3的约化的一些限制条件下，建立了椭圆曲线的奇偶猜想：E/K的Mordell-Weil秩的奇偶性与解析秩的奇偶性一致，由根数决定.证明了Q上所有椭圆曲线和素数p的p-奇偶猜想：p^ \infty -Selmer秩的奇偶性与解析秩一致.
We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott-Halberstam Conjecture, we show that all large natural numbers n with 8 not dividing n, n not congruent to 2 modulo 3, and n not congruent to 14 modulo 16, are the sum of 2 squares and 3 biquadrates.
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q…
We give some new $q$-supercongruences on truncated forms of squares of basic hypergeometric series. Most of them are modulo the cube of a cyclotomic polynomial, and two of them are modulo the fourth power of a cyclotomic polynomial. The main ingredients of our proofs are the creative microscoping method, a lemma of El Bachraoui, and the Chinese remainder theorem for coprime polynomials. We also propose several related conjectures for further study.
In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive integers $a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ax^2+by^2+cT_z$ represents sufficiently large integers and establish similar results for the forms $ax^2…
Sequentially congruent partitions and partitions into squares
本文通过构造双射证明序列一致划分与平方数划分计数相等，并推广至k次幂划分。
In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…
In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show…
On polynomials with roots modulo almost all primes
该研究定义了在几乎所有素数模下有根但无整数根的首一整系数多项式为例外多项式，分类了与不可约首一二次多项式乘积为例外多项式的不可约首一整系数多项式，并构造了指定因式形式的例外多项式。
Square-free discriminants of Frobenius rings
设$E$是$\Q$上的椭圆曲线。众所周知，$E_p$的自同态环，$E$模普通约化的素数$p$的约化，是由Frobenius元素$π_p$生成的二次虚域$Q(π_p)$的阶。当曲线有复数乘法（CM）时，这总是一个固定的字段，因为素数变化。然而，当曲线没有CM时，我们所知甚少，不仅关于阶，而且关于可能出现为改变素数的自同态代数的域。自同态环与Frobenius元的特征多项式的判别式$a_p^2-4p$的运算密切相关。在本文中，我们感兴趣的功能$π_{E,r,h}(x)$计数的素数$p$的数量$x$，使$a_p^2-4p$是无平方和在同余类$r$模$h$。本文给出了在有理数上定义的椭圆曲线上求平均时$π_{E,r,h}(x)$的精确渐近性，并讨论了这个结果与Lang-Trotter猜想以及与曲线模$p$有关的其它问题的关系.
On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
本文研究椭圆曲线二次扭曲下Birch和Swinnerton-Dyer公式模平方的有效性，通过局部Tamagawa数的分类结果，证明了满足修正Heegner假设的二次扭曲下该公式模平方的等价性，并得到半稳定椭圆曲线的相关推论。
We prove that on the cyclic groups of odd order d, there exist non zero functions whose convolution square f*f(2t) is proportional to their square f(t)^2 when the proportionality constant is given by an imaginary quadratic integer of norm d which is equal to 1 modulo 2. The proof involves theta functions on elliptic curves with complex multiplication.
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …
For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert αn + β\rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $α$ is quadratic irrational then the asymptotic formulas holds for all sufficiently large $x$. Our ingredients come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential s…
Distribution of sums of square roots modulo $1$
本文改进Steinerberger（2024）的结果，证明模1下k个平方根之和可达到O(n^{-k/2})的分布精度，并给出逼近任意实数时的指数下界，且方法避免使用指数和。
Two dimensional arithmetic progressions avoiding squares
研究了二维算术进展中避免非零平方数的结构，证明其元素数量上限，并讨论了相关下界及与素数二次非剩余下界的关系。
We prove that for all squarefree $m$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero squares the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds, where $n$ denotes the number of odd prime divisors of $m$.
E. Artin conjectured that any integer $a >1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan [10] showed that if $\sum_{p<x} \frac 1 {f_a(p)}= O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2πi/p})$ corresponding to the subgroups $<a> \subseteq \mathbb F*_p.$
Building upon the work of A. Booker and C. Pomerance (2017), we prove that for a prime power $q \geq 7$, every residue class modulo an irreducible polynomial $F \in \mathbb{F}_q[X]$ has a non-constant, square-free representative which has no irreducible factors of degree exceeding $\text{deg}~F -1$. We also give applications to generating sequences of irreducible polynomials.
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and it depends on new criteria ensuring all iterates of f are irreducible. In particular when F is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic f such that every iterate f^n is irreducible over F, but f^n is reducible modulo all primes of F for n at least 2. We also give an example for each n of a q…
We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour distances between normalised integral points on the sphere.
Does Ten Have a Friend?
任何非10的正整数$n$，其绝对指数为9/5，必须是一个至少有6个不同素因子的平方，最小的是5。此外，在$n$的素幂因子分解中，至少一个素因子必须与1模3全等，并且出现时具有与2模6全等的指数。
For positive integers $m$ we consider the theta functions $f_m(z):=\sum_{mk+1\text{ square }}q^k$. Due to classical identities of Jacobi, it is known that $$f_4\equiv f_6f_{12}\pmod 2.$$ Here we prove that the only triples $(a,b,c)$ for which $f_a\equiv f_bf_c\pmod 2$ are of the form $(2q,4q,4q)$ or $(4q,4q,8q)$, where $q$ is any positive odd number, or belong to the following finite list $$\{(4,6,12),(6,8,24),(8,12,24),(10,12,60),(15,24,40),(16,24,48),(20,24,120),(21,24,168)\}.$$ The result is inspired by the Ballantine-Merca Conjecture on recurrence relations for the parity of the partition …
The distribution of spacings between quadratic residues, II
研究了当q的素除子个数趋于无穷大时，模q的平方间距的分布。Kurlberg和Rudnick在较早的论文中证明了无平方根q的间距分布是泊松分布，本文将这一结果推广到任意q。
Transcendence of Tonelli--Shanks Power Modulo Infinitely Large Primes
该数学研究证明了无穷大素数模整数环中，与Tonelli–Shanks算法相关的多个数论对象的像在有理数域上具有超越性。
Moment Estimates and Discrepancy for Sums of Square Roots Modulo One
研究模一方根和的分布，通过二阶和四阶矩估计匹配对角尺度，并结合指数和估计与Erdős–Turán不等式，得到差异上界。
Sum of Distinct Biquadratic Residues Modulo Primes
本文证明了关于模素数下双二次多项式不同输出值之和的两个猜想，给出依赖 p 模 8 余数和 c 二次剩余性的八情形公式，并据此分类了所有可能和值的集合。
We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ can be written as a sum of three squares. Hensel's Theorem for $p$-adic integers plays a key role. Final…
The set of non-squares in a number field is diophantine
修复一个数字字段k。我们证明了k* - k*^2是k上的丢番图。这是从一个定理推导出来的，即对于k[x]中的一个非常数可分多项式P（x），在k* 模平方中至多有200个a，使得对于由y^2 - az^2 = P（x）给出的圆锥丛X的哈塞原理存在Brauer-Manin障碍。
In the present paper we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer $n$ to counting points on elliptic curves modulo $n$, succeeding with probability $1-\varepsilon$, for any $\varepsilon<0$ .
Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$ is the set of primes congruent to $r$ modulo $m$ (where $m$ and $r$ are relatively prime numbers), then this natural density is shown to be $0$.
Number-theory renormalization of vacuum energy
本文指出 d>=3 维格点 QFT 在特定动量平方 Hamiltonian 与模 N 剩余类环计算下真空能量为零，并将其联系到 d 个平方和表示数的数论问题。
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