Yves Gallot - Academia.edu (original) (raw)

Papers by Yves Gallot

arXiv (Cornell University), Sep 17, 2012

For a fixed prime p, the maximum coefficient (in absolute value) M (p) of the cyclotomic polynomi... more 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) ≤ (p + 1)/2. In 2009 Gallot and Moree showed that M (p) ≥ 2p(1 −)/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 counterexamples to Sister Beiter's conjecture and sharpen the above lower bound for M (p).

arXiv (Cornell University), Sep 16, 2022

We present an efficient proof scheme for any instance of left-to-right modular exponentiation, us... more 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 ≡ r (mod 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.

Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. ... more Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. 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> 10107. Here we achieve m> 10109 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/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. 1.

Abstract. In 1998, Eric Brier proved the existence of some numbers k such that k · 2n is never ad... more Abstract. In 1998, Eric Brier proved the existence of some numbers k such that k · 2n is never adjacent to a prime. At that time, the smallest known ”Brier Number ” was a 41-digit number. The search was extended to find the smallest Brier number. Today, the smallest known number of this form is the 27-digit number k = 878503122374924101526292469. 1. Definitions Definition 1.1. A Sierpiński number is a positive integer k such that k · 2n+1 is not prime for any integer n. Definition 1.2. A Riesel number is a positive integer k such that k · 2n − 1 is not prime for any integer n. Definition 1.3. A Brier number is both a Sierpiński number and a Riesel num-ber. 2. A constructive approach Let S = {p1, p2,..., ps} a set of prime numbers and P = ∏1≤i≤s pi. Let ei the order of 2 modulo pi (see [1, Definition 22]) and eS = lcm(e1, e2,..., es). Definition 2.1. If one of the primes of the set S divides k · 2n+1 for any number n, and if for every prime pi of S there is at least one ni such tha...

When we search for prime numbers in a sequence, we would like to estimate how many are prime in a... more When we search for prime numbers in a sequence, we would like to estimate how many are prime in a fixed range with a simple method before starting a long computation. Today, our mathematical knowledge in this domain is about zero. Only the problem of the sequences of the form a · n + b was solved by Hadamard

Abstract. Is it possible to improve the convergence properties of the series for the computation ... more Abstract. Is it possible to improve the convergence properties of the series for the computation of the Cn involved in the distribution of the generalized Fermat prime numbers? If the answer to this question is yes, then the search for a large prime number P will be C · log(P) times faster than today, where C ≈ 0.01. 1.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

If the equation of the title has an integer solution with k>2, then m>10^9.3·10^6. This was... more If the equation of the title has an integer solution with k>2, then m>10^9.3·10^6. 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 2 and making an extensive continued fraction digits calculation of (2)/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.

arXiv: Number Theory, 2008

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the... more 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.

The sequence of numbers generated by the cyclotomic polynomials Φn(2) contains the Mersenne numbe... more The sequence of numbers generated by the cyclotomic polynomials Φn(2) contains the Mersenne numbers 2p − 1 and the Fermat numbers 22 m + 1. Does an algorithm involving O(n) modular operations exist to test the primality of Φn(b)? 1. Cyclotomic polynomials Let n be a positive integer and let ζn be the complex number e2πi/n. The nth cyclotomic polynomial is, by definition

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

Journal of Open Research Software, 2015

Genefer is a suite of programs for performing Probable Primality (PRP) tests of Generalised Ferma... more Genefer is a suite of programs for performing Probable Primality (PRP) tests of Generalised Fermat numbers b 2 n +1 (GFNs) using a Fermat test. Optimised implementations are available for modern CPUs using single instruction, multiple data (SIMD) instructions, as well as for GPUs using CUDA or OpenCL. Genefer has been extensively used by PrimeGrid-a volunteer computing project searching for large prime numbers of various kinds, including GFNs. Genefer's architecture separates the high level logic such as checkpointing and user interface from the architecture-specific performance-critical parts of the implementation, which are suitable for re-use. Genefer is released under the MIT license. Source and binaries are available from www.assembla.com/ spaces/genefer.

arXiv: Number Theory, 2008

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime... more 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.

The sequence of numbers generated by the cyclotomic polynomi- alsn(2) contains the Mersenne numbe... more The sequence of numbers generated by the cyclotomic polynomi- alsn(2) contains the Mersenne numbers 2p Ä 1 and the Fermat numbers 22 m + 1. Does an algorithm involving O(n) modular operations exist to test the primality ofn(b)?

Is it possible to improve the convergence properties of the series for the computation of the Cn ... more Is it possible to improve the convergence properties of the series for the computation of the Cn involved in the distribution of the generalized Fermat prime numbers? If the answer to this question is yes, then the search for a large prime number P will be C Å log(P) times faster than today, where C ô 0:01.

arXiv (Cornell University), Sep 17, 2012

For a fixed prime p, the maximum coefficient (in absolute value) M (p) of the cyclotomic polynomi... more 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) ≤ (p + 1)/2. In 2009 Gallot and Moree showed that M (p) ≥ 2p(1 −)/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 counterexamples to Sister Beiter's conjecture and sharpen the above lower bound for M (p).

arXiv (Cornell University), Sep 16, 2022

We present an efficient proof scheme for any instance of left-to-right modular exponentiation, us... more 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 ≡ r (mod 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.

Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. ... more Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. 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> 10107. Here we achieve m> 10109 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/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. 1.

Abstract. In 1998, Eric Brier proved the existence of some numbers k such that k · 2n is never ad... more Abstract. In 1998, Eric Brier proved the existence of some numbers k such that k · 2n is never adjacent to a prime. At that time, the smallest known ”Brier Number ” was a 41-digit number. The search was extended to find the smallest Brier number. Today, the smallest known number of this form is the 27-digit number k = 878503122374924101526292469. 1. Definitions Definition 1.1. A Sierpiński number is a positive integer k such that k · 2n+1 is not prime for any integer n. Definition 1.2. A Riesel number is a positive integer k such that k · 2n − 1 is not prime for any integer n. Definition 1.3. A Brier number is both a Sierpiński number and a Riesel num-ber. 2. A constructive approach Let S = {p1, p2,..., ps} a set of prime numbers and P = ∏1≤i≤s pi. Let ei the order of 2 modulo pi (see [1, Definition 22]) and eS = lcm(e1, e2,..., es). Definition 2.1. If one of the primes of the set S divides k · 2n+1 for any number n, and if for every prime pi of S there is at least one ni such tha...

When we search for prime numbers in a sequence, we would like to estimate how many are prime in a... more When we search for prime numbers in a sequence, we would like to estimate how many are prime in a fixed range with a simple method before starting a long computation. Today, our mathematical knowledge in this domain is about zero. Only the problem of the sequences of the form a · n + b was solved by Hadamard

Abstract. Is it possible to improve the convergence properties of the series for the computation ... more Abstract. Is it possible to improve the convergence properties of the series for the computation of the Cn involved in the distribution of the generalized Fermat prime numbers? If the answer to this question is yes, then the search for a large prime number P will be C · log(P) times faster than today, where C ≈ 0.01. 1.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

If the equation of the title has an integer solution with k>2, then m>10^9.3·10^6. This was... more If the equation of the title has an integer solution with k>2, then m>10^9.3·10^6. 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 2 and making an extensive continued fraction digits calculation of (2)/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.

arXiv: Number Theory, 2008

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the... more 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.

The sequence of numbers generated by the cyclotomic polynomials Φn(2) contains the Mersenne numbe... more The sequence of numbers generated by the cyclotomic polynomials Φn(2) contains the Mersenne numbers 2p − 1 and the Fermat numbers 22 m + 1. Does an algorithm involving O(n) modular operations exist to test the primality of Φn(b)? 1. Cyclotomic polynomials Let n be a positive integer and let ζn be the complex number e2πi/n. The nth cyclotomic polynomial is, by definition

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

Journal of Open Research Software, 2015

Genefer is a suite of programs for performing Probable Primality (PRP) tests of Generalised Ferma... more Genefer is a suite of programs for performing Probable Primality (PRP) tests of Generalised Fermat numbers b 2 n +1 (GFNs) using a Fermat test. Optimised implementations are available for modern CPUs using single instruction, multiple data (SIMD) instructions, as well as for GPUs using CUDA or OpenCL. Genefer has been extensively used by PrimeGrid-a volunteer computing project searching for large prime numbers of various kinds, including GFNs. Genefer's architecture separates the high level logic such as checkpointing and user interface from the architecture-specific performance-critical parts of the implementation, which are suitable for re-use. Genefer is released under the MIT license. Source and binaries are available from www.assembla.com/ spaces/genefer.

arXiv: Number Theory, 2008

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime... more 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.

The sequence of numbers generated by the cyclotomic polynomi- alsn(2) contains the Mersenne numbe... more The sequence of numbers generated by the cyclotomic polynomi- alsn(2) contains the Mersenne numbers 2p Ä 1 and the Fermat numbers 22 m + 1. Does an algorithm involving O(n) modular operations exist to test the primality ofn(b)?

Is it possible to improve the convergence properties of the series for the computation of the Cn ... more Is it possible to improve the convergence properties of the series for the computation of the Cn involved in the distribution of the generalized Fermat prime numbers? If the answer to this question is yes, then the search for a large prime number P will be C Å log(P) times faster than today, where C ô 0:01.