arXiv subjects

François Wolf

Publications and source records attributed to François Wolf.

5 recordsLinked to original sources

arXiv2024

Semi-primitive roots and irreducible quadratic forms

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any GCD calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [4].

math.GM↗

arXiv2023

On the density of primes of the form $X^2+c$

We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating values of $c$ such that a given prime $p$ is the minimum of the union of prime divisors of all elements in $E_c$. We also present quadratic forms generating divisors of Ec and study the prime divisors of its terms. This paper uses the results of Dirichlet's arithmetic progression theorem [1] and the article [6] to rewrite a conjecture of Shanks [2] on the density of primes in $E_c$. Finally, based on these results, we discuss the heuristics of large primes occurrences in the research set of our algorithm.

cs.DS↗

arXiv2022

On the factorization of numbers of the form $X^2+c$

We study the factorization of the numbers $N = X^2+c$, where $c$ is a fixed constant, and this independently of the value of gcd$(X,c)$. We prove the existence of a family of sequences with arithmetic difference $(U_n, Z_n)$ generating factorizations, i.e. such that: $(U_n)^2+c = Z_nZ_{n+1}$. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.

math.GM↗

arXiv2021

Primality test and primes enumeration using odd numbers indexation

Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves computational time by a constant. We then apply similar techniques to Atkin prime-numbers sieve which uses modulus operations and finally to Pritchard wheel sieve, in both case yielding similar results.

math.GM↗

arXiv2018

Calculation of extended gcd by normalization

We propose a new algorithm solving the extended gcd problem, which provides a solution minimizing one of the two coordinates. The algorithm relies on elementary arithmetic properties.

cs.DS↗