Related papers: Mersenne numbers and the doubling map
We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…
An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an…
The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set…
We exhibit twelve new primitive trinomials over GF(2) of record degrees $42643801$, $43112609$, and $74207281$. In addition we report the first Mersenne exponent not ruled out by Swan's theorem - namely $57885161$ - for which no primitive…
The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their…
We present a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a co-prime pair $(m,n)$ of positive integers. This work is inspired by Thomas-Williams work on the modular sweep map. A simple proof of the…
In his 1963 paper on the sum of consecutive primes, Moser posed four open questions related to $f(n)$, the number of ways an integer $n$ can be written as a sum of consecutive primes. (See also problem C2 from Richard K.~Guy's…
Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…
In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers…
We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…
The main purpose of this paper is pedagogical. Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations…
The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$…
This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only…
In this article, I study and solve the exponential Diophantine equation $M_p^{x} + (M_q + 1)^{y}= (lz)^2$ where $M_p$ and $M_q$ are Mersenne primes, $l$ is a prime number, and $x,y$, and $z$ are non-negative integers. Several illustrations…
A discrete map based on the sum of an integer's distinct primes factors and the sum of its other factors is defined and its iteration is studied.
We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm…
We present two modified versions of the primal-dual splitting algorithm relying on forward-backward splitting proposed in \cite{vu} for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators…
Multiple Deligne values (MDVs) are iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$ and $D=-d\log(1-\lambda x)$, where $\lambda$ is a primitive sixth root of unity. MDVs of weight 11…
We show that the product of an nx3 matrix and a 3x3 matrix over a commutative ring can be computed using 6n+3 multiplications. For two 3x3 matrices this gives us an algorithm using 21 multiplications. This is an improvement with respect to…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…