arXiv subjects

Yves Gallot

Publications and source records attributed to Yves Gallot.

8 recordsLinked to original sources

arXiv2023

An Efficient Modular Exponentiation Proof Scheme

We present an efficient proof scheme for any instance of left-to-right modular exponentiation, used in many computational tests for primality. Specifically, we show that for any $(a,n,r,m)$ the correctness of a computation $a^n\equiv r\pmod m$ can be proven and verified with an overhead negligible compared to the computational cost of the exponentiation. Our work generalizes the Gerbicz-Pietrzak proof scheme used when $n$ is a power of $2$, and has been successfully implemented at PrimeGrid, doubling the efficiency of distributed searches for primes.

cs.CR↗

arXiv2015

Addendum to the paper: "Artin Prime Producing Quadratics" [Abh. Math. Sem. Univ. Hamburg 77 (2007), 109--127; MR2379332 (2008m:11194)] by P. Moree

Can one find an integer $g$ and a polynomial $f$, such that $g$ is a primitive root for many consecutive (different) prime values assumed by $f$? Moree considered this problem in 2007 with computational assistence from Gallot and concentrated on the case where $f$ is quadratic. Recently Akbary and Scholten (Math. Comp., to appear) extended this work and also considered the case where $f$ is linear and cubic. In their paper they improved on a record of the authors in the quadratic case. However, mistakingly Moree had not mentioned in the 2007 paper the then record (from 2006), but only an older record. The 2006 record (due to Gallot) actually exceeds the 'record' from Akbary and Scholten. Aside from pointing this out, the goal of the note is to attend people to the problem mentioned in the first sentence of the abstract and point out some connections with recent spectacular work of Maynard and highly interesting work of Pollack.

math.NT↗

arXiv2012

Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses

For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ of the cyclotomic polynomial $Φ_{pqr}(x)$, where $r$ and $q$ are free primes satisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that $M(p)\le(p+1)/2$. In 2009 Gallot and Moree showed that $M(p)\ge 2p(1-ε)/3$ for every $p$ sufficiently large. In this article Kloosterman sums (`cloister man sums') and other tools from the distribution of modular inverses are applied to quantify the abundancy of counter-examples to Sister Beiter's conjecture and sharpen the above lower bound for $M(p)$.

math.NT↗

arXiv2011

The family of ternary cyclotomic polynomials with one free prime

A cyclotomic polynomial Φ_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ_{pqr}(x) with p<q<r, p and q fixed and r a free prime.

math.NT↗

arXiv2009

The Erd\H{o}s--Moser equation $1^k+2^k+...+(m-1)^k=m^k$ revisited using continued fractions

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.

math.NT↗

arXiv2008

Neighboring ternary cyclotomic coefficients differ by at most one

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.

math.NT↗

arXiv2008

Ternary cyclotomic polynomials having a large coefficient

Let $Φ_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $Φ_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p 0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1 (2/3-ε)p_j$ for $j\ge 1$.

math.NT↗

arXiv2008

Value distribution of cyclotomic polynomial coefficients

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we study the average of the a_n(k). We derive analogous results for the kth Taylor coefficient of 1/Phi_n(x) (taken around x=0), the kth coefficient of the nth reciprocal cyclotomic polynomial. We formulate various open problems.

math.NT↗