Fermat and Mersenne numbers in $k$-Pell sequence

For an integer $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence, which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is defined by the recurrence$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)},\quad \text{for all }n \geq 2.$For any positive integer $n$, a number of the form $2^n+1$ is referred to as a Fermat number, while a number of the form $2^n-1$ is referred to as a Mersenne number. The goal of this paper is to determine Fermat and Mersenne numbers which are members of the $k$-generalized Pell sequence. More precisely, we solve the Diophantine equation $P^{(k)}_n=2^a\pm 1$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$. We prove a theorem which asserts that, if the Diophantine equation $P^{(k)}_n=2^a\pm 1$ has a solution $(n,a,k)$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$, then we must have that $(n,a,k)\in \{(1,1,k),(3,2,k),(5,5,3)\}$. As a result of our theorem, we deduce that the number $1$ is the only Mersenne number and the number $5$ is the only Fermat number in the $k$-Pell sequence.

Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software

In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors.

An open source software package for primality testing of numbers of the form p2^n+1, with no constraints on the relative sizes of p and 2^n

We develop an efficient software package to test for the primality of p2^n+1, p prime and p>2^n. This aids in the determination of large, non-Sierpinski numbers p, for prime p, and in cryptography. It furthermore uniquely allows for the computation of the smallest n such that p2^n+1 is prime when p is large. We compute primes of this form for the first one million primes p and find four primes of the form above 1000 digits. The software may also be used to test whether p2^n+1 divides a generalized fermat number base 3.

An open source software package for primality testing of numbers of the form p2^n+1, with no constraints on the relative sizes of p and 2^n

We develop an efficient software package to test for the primality of p2^n+1, p prime and p>2^n. This aids in the determination of large, non-Sierpinski numbers p, for prime p, and in cryptography. It furthermore uniquely allows for the computation of the smallest n such that p2^n+1 is prime when p is large. We compute primes of this form for the first one million primes p and find four primes of the form above 1000 digits. The software may also be used to test whether p2^n+1 divides a generalized fermat number base 3.