EXTREMELY FASCINATING Property of Mersenne Numbers

2002

Glasnik Matematicki, 2016

For an integer k ≥ 2, let (F (k) n )n be the k-Fibonacci sequence which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all k-Fibonacci numbers which are Mersenne numbers, i.e., k-Fibonacci numbers that are equal to 1 less than a power of 2. As a consequence, for each fixed k, we prove that there is at most one Mersenne prime in (F

Indonesian Journal of Mathematics Education, 2020

Mersenne primes are a specific type of prime number that can be derived using the formula M_p=2^p - 1, where p is a prime number. A perfect number is a positive integer of the form P(p)=2^(p-1)(2^p - 1) where 2^p - 1 is a Mersenne prime and can be written as the sum of its proper divisor, that is, a number which is half the sum of all of its positive divisor. In this paper, some concepts relating to Mersenne primes and perfect numbers were revisited. Mersenne primes and perfect numbers were evaluated using triangular numbers. Further, this paper discussed how to partition perfect numbers into odd cubes for odd prime The formula that partition perfect numbers in terms of its proper divisors were developed. The results of this study are useful to understand the mathematical structures of Mersenne primes and perfect numbers.

Notes on Number Theory and Discrete Mathematics, 2021

In this paper, we determine all the Mersenne numbers which are in the sequences of Padovan and Perrin numbers, respectively.

Mersenne Variant Numbers and Integers investigated for Primality, Factorization, 2008

Mersenne Variants are numbers of the form s^n +/- c where s,n >=2 and -s-1<=c<=s+1 and gcd(s,c)=1 which is a generalization of the type of Fermat and Mersenne Numbers. Here in this paper we give algorithms for primality of such numbers with mathematical proofs of correctness.

Italian Journal of Pure and Applied Mathematics, 2024

This research work focused on studying an exponential Diophantine equation involving Mersenne numbers. Specifically, it sought to find the nonnegative integer solutions (Mn, x, y, z) of the Diophantine equation (Mn)^x + (Mn + 1)^y = z^2. To obtain the solutions, a combination of modular arithmetic method and factoring method, together with some other results like Mihailescu’s theorem, was utilized. Results of the Diophantine analysis revealed that aside from (3, 2, 2, 5) and (7, 0, 1, 3), the equation has infinitely many solutions of the form (2^2k −1, 1, 0, 2^k), where k is a positive integer.

Matematychni Studii, 2021

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...

Lecture Notes in Computer Science, 2006

arXiv (Cornell University), 2021

In this paper, we define generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and the relation with classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.

Mersenne Variants, Primality, Factorization, 2014

Mersenne Variants are numbers of the form $s^n \pm c$ where s,n,c are numbers. s>=2, n>=2, -s-1 <= c <= s+1. and the GCD(s,c)=1. The paper contains several primality tests of Mersenne Variants and proofs of correctness of the Algorithms based on Modified Lucas-Lehmer Tests.