Mersenne Numbers and Fermat Numbers

10.1142/12100 ◽  
2021 ◽  
Author(s):  
Elena Deza
Keyword(s):  
Fermat Numbers ◽  
Mersenne Numbers

scholarly journals Fermat numbers and Mersenne numbers

1964 ◽  
Vol 18 (85) ◽  
pp. 146-146 ◽  
Author(s):  
J. L. Selfridge ◽  
Alexander Hurwitz
Keyword(s):  
Fermat Numbers ◽  
Mersenne Numbers

scholarly journals Prime Fermat numbers and maximum determinant matrix conjecture

2020 ◽  
pp. 2-9
Author(s):  
Nikolay Balonin ◽  
Mikhail Sergeev ◽  
Anton Vostrikov
Keyword(s):  
Simple Structure ◽  
Close Relation ◽  
Extremal Problems ◽  
Matrix Elements ◽  
Video Information ◽  
Fermat Numbers ◽  
Mersenne Numbers ◽  
Precise Evaluation ◽  
The Matrix ◽  
Extreme Solutions

Purpose: Solution to the problem of optimizing the determinants of matrices with a modulus of entries < 1. Developing a theory of such matrices based on preliminary research results. Methods: Extreme solutions (in terms of the determinant) are found by minimizing the absolute values of orthogonal matrix elements, and their subsequent classification. Results: Matrices of orders equal to prime Fermat numbers have been found. They are special, as their absolute determinant maximums can be reached on a simple structure. We provide a precise evaluation of the determinant maximum for these matrices and formulate a conjecture about it. We discuss the close relation between the solutions of extremal problems with the limitation on the matrix column orthogonality and without it. It has been shown that relative maximums of orthogonality-limited matrix determinants correspond to absolute maximums of orthogonality-unlimited matrix determinants. We also discuss the ways to build extremal matrix families for the orders equal to Mersenne numbers. Practical relevance: Maximum determinant matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for maximum determinant matrix search and a library of constructed matrices are used in the mathematical network “mathscinet.ru” along with executable online algorithms.

scholarly journals The q-Integers and the Mersenne Numbers

2018 ◽  
Author(s):  
Amelia Carolina Sparavigna
Keyword(s):  
Mersenne Numbers

A Report on Primes of the Form k⋅2 n + 1 and On Factors of Fermat Numbers

1958 ◽  
Vol 9 (5) ◽  
pp. 673 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Fermat Numbers

scholarly journals Note on Mersenne numbers

1932 ◽  
Vol 38 (6) ◽  
pp. 383-385 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Mersenne Numbers

scholarly journals Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software

2021 ◽  
pp. 167-173
Author(s):  
Jianhui Li ◽  
◽  
Manlan Liu
Keyword(s):  
Parallel Computing ◽  
Parallel Algorithm ◽  
Computational Efficiency ◽  
Sequential Algorithm ◽  
Parallel Processes ◽  
Fermat Numbers ◽  
Fermat Number ◽  
Parallel Strategy

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.

scholarly journals New factors of Mersenne numbers

1961 ◽  
Vol 15 (73) ◽  
pp. 51-51 ◽  
Author(s):  
Edgar Karst
Keyword(s):  
Mersenne Numbers

scholarly journals Four new factors of Fermat numbers

1978 ◽  
Vol 32 (143) ◽  
pp. 941 ◽  
Author(s):  
D. E. Shippee
Keyword(s):  
Fermat Numbers

scholarly journals On the square-freeness of Fermat and Mersenne numbers

1967 ◽  
Vol 22 (3) ◽  
pp. 563-564 ◽  
Author(s):  
Leroy Warren ◽  
Henry Bray
Keyword(s):  
Mersenne Numbers
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