Ternary cyclotomic polynomials having a large coefficient

Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\ge 11$. We also prove that given any $\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<...$ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\epsilon)p_j$ for $j\ge 1$.Comment: 19 pages, 6 tables, to appear in Crelle's Journal. Revised version with many small change

The Erd\H{o}s--Moser equation 1k+2k+...+(m−1)k=mk1^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.Comment: 17 page

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 $\Phi_{pqr}(x)$, 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 $M(p)\ge 2p(1-\epsilon)/3$ 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 $M(p)$.Comment: 2 figures; 15 page

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.Comment: 11 pages, 2 table

An Efficient Modular Exponentiation Proof Scheme

We present an efficient proof scheme for any instance of left-to-right modular exponentiation, used in the Fermat probable prime test. Specifically, we show that for any $(a,n,r,m)$ the claim $a^n\equiv r\pmod m$ can be proven and verified with an overhead negligible compared to the computational cost of the exponentiation. Our work generalizes the Gerbicz-Pietrzak double check scheme, greatly improving the efficiency of general probabilistic primality tests in distributed searches for primes such as PrimeGrid

The family of ternary cyclotomic polynomials with one free prime

A cyclotomic polynomial \Phi_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 \Phi_{pqr}(x) with p<q<r, p and q fixed and r a free prime.Comment: To appear in Involve, 23 pages, 7 Tables, Question 5 has been meanwhile answered in the positive by Eugenia Rosu; (extended version, 32 pages

Anthropocène : Plan B, création de connaissances pour répondre aux enjeux sociétaux de manière soutenable dans les limites planétaires

De nombreuses recherches et en particulier celles sur les limites planétaires ont montré que nous dépassons actuellement plusieurs limites globales, ce qui questionne fortement la soutenabilité de nos sociétés contemporaines à forte empreinte écologique. Cette prise de conscience se généralise et a fait croître à une vitesse importante les attentes sociétales de visions alternatives à un futur basé sur le seul progrès technologique et/ou une croissance économique infinie.Nous souhaitons faire face à ces constats et aux attentes qu’ils génèrent, sans greenwashing et sans nous en remettre à une croissance verte que nous savons impossible depuis longtemps et notamment par les travaux commandités par le Club de Rome. Impossibilité qui a été rappelée récemment à notre mémoire par une note de l’UE. Pour cela nous souhaitons engager l’UGA dans la construction, sur le long terme, d’une communauté scientifique transdisciplinaire. Nous proposons de développer des recherches complémentaires et alternatives à celles basées sur la double hypothèse d’un éternel progrès technologique et d’une croissance économique qui serait nécessairement vertueuse sur le plan social. Ces recherches auront pour objectif principal d’appréhender la dimension systémique et complexe des questions de dépassement écologique.Pour cela nous savons que nous pouvons d’ores et déjà appuyer notre démarche sur plusieurs collectifs de personnels et d’étudiants nés spontanément dans différentes structures de l’UGA. Leur diversité de profils et de disciplines constitue un atout précieux pour construire une approche transdisciplinaire. Nous pensons donc qu’il est utile et pertinent d’essayer de fédérer ces initiatives dans une démarche collective commune de production de connaissances