Bartosz Zralek
We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo $p$ of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if $p-1$ has a $(\ln p)^{O(1)}$-smooth divisor exceeding $(p-1)^{{1/2}+δ}$ for some arbitrary small $δ$. We also address the issue of computing roots modulo $p$ in deterministic time.
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Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
In this paper we will consider new bounds on the smallest primitive root modulo a prime. We will make more judicious use of the Pòlya--Vinogradov and Burgess inequalities, and use them to prove that the smallest primitive root is smaller than $p^{0.68}$ for all primes $p$.
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 provide an explicit estimate on the least primitive root mod $p^{2}$. We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$.
For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.
A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the know…
A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.
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。
Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.
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。
Artin's Conjecture on Primitive Roots states that a non-square nonunit integer $a$ is a primitive root modulo $p$ for the positive proportion of $p$. This conjecture remains open, but on average, there are many results due to P. J. Stephens. There is a natural generalization of the conjecture for composite moduli. We can consider $a$ as the primitive root modulo, $(\mathbb{Z}/n\mathbb{Z})^{*}$ if $a$ is an element of the maximal exponent in the group. The behavior is more complex for composite moduli, and the corresponding average results are provided by S. Li and C. Pomerance, and recently by…
We establish a connection between semi-primitive roots of the multiplicative group of integers modulo $2^{k}$ where $k\geq 3$, and the logarithmic base in the algorithm introduced by Fit-Florea and Matula (2004) for computing the discrete logarithm modulo $2^{k}$. Fit-Florea and Matula used properties of the semi-primitive root 3 modulo $2^{k}$ to obtain their results and provided a conversion formula for other possible bases. We show that their results can be extended to any semi-primitive root modulo $2^{k}$ and also present a generalized version of their algorithm to find the discrete logar…
We prove that any prime $p$ satisfying $ϕ(p-1)\leq (p-1)/4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$.
A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Ch…
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.
Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is conjecturally the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.
The Hilbert class polynomial $H_{\mathcal{O}}(x)\in \mathbb{Z}[x]$ attached to an order $\mathcal{O}$ in an imaginary quadratic field $K$ is the monic polynomial whose roots are precisely the distinct $j$-invariants of elliptic curves over $\mathbb{C}$ with complex multiplication by $\mathcal{O}$. Let $p$ be a prime inert in $K$ and strictly greater than $|\operatorname{disc}(\mathcal{O})|$. We show that the number of $\mathbb{F}_p$-roots of $H_\mathcal{O}(x)\!\! \pmod{p}$ is either zero or $|\operatorname{Pic}(\mathcal{O})[2]|$ by exhibiting a free and transitive action of $\operatorname{Pic}…
$ $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 …
Polynomials with factors of the form $(x^q-a)$ with roots modulo every integer
本文给出了多项式 $\prod_{j=1}^{l} (x^{q} - a_{j})$ 对每个正整数均有根的充要条件，并证明了在 $l \leq q$ 且无完全 $q$ 次幂因子时存在无根情况，同时建立了基于 $q$-free 部分的等价判据。
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.
We study the difference between the number of primitive roots modulo $p$ and modulo $p+k$ for prime pairs $p,p+k$. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs $p,p+k$, the dominant inequality is reversed.
We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime $p$, which by the well-known result of Burgess are known to be at most $p^{1/4+o(1)}$. Here we measure the distance in the Hamming metric and show that if $p$ is a sufficiently large $r$-bit prime, then for any integer $n \in [1,p]$ one can obtain a primitive root modulo $p$ by changing at most $0.11002786...r$ binary digits of $n$. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very sm…
The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. Similar to Duke, Friedlander and Iwaniec, w…
Conjectures on the distribution of roots modulo a prime of a polynomial
本文基于数值实验提出关于首一整系数多项式模完全可分解素数时局部根归一化分布及带同余约束根分布的两个新猜想，后者在 n=1 时退化为 Dirichlet 算术级数定理。
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 $<a> \subseteq \mathbb F*_p.$
Galois representations modulo $p$ that do not lift modulo $p^2$
本文确定了具有足够单位根域上 $H^2(H,A)$ 中的可忽略类子群，并据此构造了不能模 $p^2$ 提升的连续 $n$ 维模 $p$ Galois 表示，回答了 Khare 和 Serre 的问题并推翻了 Florence 的猜想。
Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f(x) and g(x) are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence (u/n,v/n), with f(u) = 0(n) and g(v) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
Computation of the least primitive root
本文计算了所有素数 $p\le 10^{16}$ 的模 $p$ 和模 $p^2$ 的最小原根，并基于计算结果证明了三个定理，特别是对所有 $p>3$ 证明 $g(p)<p^{5/8}$，对所有 $p$ 证明 $h(p)<p^{2/3}$。
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of residue classes. Using it, we are able to give an elegant description of some results due to Carl Schick, others are reduced to well-known theorems from algebra and number theory. Many concepts from number theory such as quadratic residues and primitive roots are equally applicable. It brings noticeable advantages in studying powers of odd primes, and in partic…
Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $α_1,\dots,α_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study linear relations group-theoretically. If $\mathbb{Q}(α_1)$ is an abelian extension, we can understand the structure completely.
In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.
Wooley ({\em J. Number Theory}, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the proof to a slightly different setting. Specifically, we consider polynomials with coefficients from a polynomial ring $\mathbb{F}[t]$ for an arbitrary field $\mathbb{F}$ and give an upper bound on the number of isolated roots modulo $t^s$ for an arbitrary positive integer $s$. In particular, using $s=1$, we can bound the number of isolated roots of a system of…
Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409<p< 2.5\times 10^{15}$ and for all $p>3.67\times 10^{71}$.
Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $ε> 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-ε}$. It is an open problem to show the same result with $\frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{\frac{1}{2}+\frac{1}{30}}$. The same method allows one to produce, for each $ε> 0$, explicit finite sets $\mathcal{A}$ with the property that for almost …
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by truncating certain power series with rational coefficients that satisfy simple differential equations.
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。
Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots, and (iii) every proper divisor of $f_{n}(x)$ fails to have root modulo some positive integer. We exhibit a process to explicitly construct such $f_{n}$ and this process demonstrates that the set of natural numbers $a_{n}$, such that the polynomial $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ satisfies the properties (i), (ii) and (iii), is of positive asympto…
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。
Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an application we give a sufficient condition for a prime $ p $ such that a non square number $ n $ is a primitive root modulo $ p $.
A note on least primitive roots
本文针对Grosswald关于素数最小原根的猜想，证明当素数p大于1.77×10^54时，最小原根小于√p减2，推进了该猜想的研究。
For a positive integer $N$, we define the N-rank of a non singular integer $d\times d$ matrix $A$ to be the maximum integer $r$ such that there exists a minor of order $r$ whose determinant is not divisible by $N$. Given a positive integer $r$, we study the growth of the minumum integer $k$, such that $A^k-I$ has N-rank at most $r$, as a function of $N$. We show that this integer $k$ goes to infinity faster than $\log N$ if and only if for every eigenvalue $λ$ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent…
Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $ε>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+ε} \nonumber \end{align} holds for most prime $p$ such that $p-1$ does not have small prime factors, but $2$. We also give an explicit description of the exceptional set.
On a polynomial involving quadratic residues modulo primes
本文通过伽罗瓦理论研究多项式G_p(x)在单位根处的值，验证了某些猜想，特别是针对特定素数模数的根单位情况，给出了多项式值的表达式。
The concept of p-ordering for a prime p was introduced by Manjul Bhargava (in his PhD thesis) to develop a generalized factorial function over an arbitrary subset of integers. This notion of p-ordering provides a representation of polynomials modulo prime powers, and has been used to prove properties of roots sets modulo prime powers. We focus on the complexity of finding a p-ordering given a prime p, an exponent k and a subset of integers modulo p^k. Our first algorithm gives a p-ordering for set of size n in time O(nk\log p), where set is considered modulo p^k. The subsets modulo p^k can be …
We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.
Bernstein-Sato theory modulo $p^m$
本文发展了系数在 $\mathbb{Z}/p^m$ 中的多项式的 Bernstein-Sato 多项式理论，证明了其根为有理数且负根与模 $p$ 约化一致，并发现根可能为正，同时通过 $p$-挠定义了根的“强度”，表明“强”根通过模 $p$ 约化给出特征零的根。
Let $p=2n+1$ be an odd prime, and let $ζ_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[ζ_{p^2-1}]$ be a primitive root modulo $p\mathbb{Z}_p[ζ_{p^2-1}]$ with $g\equiv ζ_{p^2-1}\pmod {p\mathbb{Z}_p[ζ_{p^2-1}]}$. Let $Δ\equiv3\pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $\mathbb{Z}$. By the Local Existence Theorem we know that $\mathbb{Q}_p(\sqrtΔ)=\mathbb{Q}_p(ζ_{p^2-1})$. For all $x\in\mathbb{Z}[\sqrtΔ]$ and $y\in\mathbb{Z}_p[ζ_{p^2-1}]$ we use $\bar{x}$ and $\bar{y}$ to denote th…
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。
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 a…
Let $p>2$ be prime and $g$ a primitive root modulo $p$. We present an argument for the fact that discrete logarithms of the numbers in any arithmetic progression are uniformly distributed in $[1,p]$ and raise some questions on the subject.
Faster Deterministic Integer Root Finding for Integer Polynomials
该研究提出一种更快的确定性整数求根算法，通过优化 $p$-进框架及候选验证算法，将无平方因子整数多项式求根的时间复杂度从 $\tilde{O}(n^2b)$ 改进为 $\tilde{O}(n^{3/2}b)$，实现了亚二次级的效率提升。
Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a root modulo n of the Hilbert class polynomial H(X) for a fundamental discriminant D. The usual way is to compute H(X) over the integers and then to find the root modulo n. We present a modified version of the Chinese remainder theorem (CRT) to compute H(X) modulo n directly fr…
A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.
We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our formulas for the $M_2$-rank at roots of unity lead to asymptotics for certain combinations of $N2(r,m,n)$ (the number of partitions without repeated odd parts of $n$ with $M_2$-rank congruent to $r$ modulo $m$). This allows us to deduce inequalities among certain combinations of $N2(r,m,n)$. In particular, we resolve a few conjectured inequalities of Mao.
An analogue of Girstmair's formula in function fields
本文研究函数域中 Girstmair 公式的类似物，建立了扩域的除子类数正负部分与 $1/P$ 在 $G$ 进制下数字之间的关系。
Compressed sensing is a celebrated framework in signal processing and has many practical applications. One of challenging problems in compressed sensing is to construct deterministic matrices having restricted isometry property (RIP). So far, there are only a few publications providing deterministic RIP matrices beating the square-root bottleneck on the sparsity level. In this paper, we investigate RIP of certain matrices defined by higher power residues modulo primes. Moreover, we prove that the widely-believed generalized Paley graph conjecture implies that these matrices have RIP breaking t…
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