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- Title
Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems.
- Authors
Pletser, Vladimir
- Abstract
A new generalized definition of Mersenne numbers is proposed of the form a n − a − 1 n , called global generalized Mersenne numbers and noted G M a , n with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, G M a , n − 1 is even and divisible by n, a and a − 1 for any prime n > 2 , and by a a − 1 + 1 for any prime n > 5 . The remaining factor is a function of triangular numbers of a − 1 , specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that G M a , n ≡ 1 or 7 mod 12 depending on the congruence of a mod 4 ; second, that G M a , n − 1 are divisible by 10 if n ≡ 1 mod 4 and, if n ≡ 3 mod 4 , G M a , n ≡ 1 or 7 or 9 mod 10 , depending on the congruence of a mod 5 ; third, that all factors c i of G M a , n are of the form 2 n f i + 1 such that c i is either prime or the product of primes of the form 2 n j + 1 , with f i , j natural integers; fourth, that for prime n > 2 , all G M a , n are periodically congruent to ± 1 or ± 3 mod 8 depending on the congruence of a mod 8 ; and fifth, that the factors of a composite G M a , n are of the form 2 n f i + 1 with f i ≡ u mod 4 with u = 0 , 1, 2 or 3 depending on the congruences of n mod 4 and of a mod 8 . The potential use of generalized Mersenne primes in cryptography is shortly addressed.
- Subjects
GEOMETRIC congruences; DEFINITIONS; INTEGERS; EXPONENTS; CRYPTOGRAPHY
- Publication
Symmetry (20738994), 2024, Vol 16, Issue 5, p551
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym16050551