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A Note on the Decimal Expansion of Reciprocals of Mersenne Primes

The aim of this note is to study, in the realm of Midy's theorem, arithmetic properties of digital blocks in the base-10 expansion of \(1 / M_p\), with \(M_p=2^p-1\) a Mersenne prime. If \(p\) divides the period length \(L\) of \(1/M_p\), one can partition the periodic part into \(p\) blocks, each of length \(\ell\).
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Mersenne prime's inducement

International Journal of Algebra, 2018
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On some geometry of Mersenne primes

Periodica Mathematica Hungarica, 1994
A possible connection between Mersenne primes and certain geometrical structures is implied. Here the authors consider the structures \((\mathbb{Z}_ q,{\mathcal B}_ p^ \#, \in)\) resulting from a planar nearring \((\mathbb{Z}_ q, +, *)\), where \(q= M_ p\) is a Mersenne prime, \(\mathbb{Z}_ q\) denotes the integers modulo \(q\), \(*\) is a ...
Clay, J. R., Yeh, Y.-N.
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The other side of Mersenne Primes [PDF]

Mersenne numbers, (2^n)-1, with natural number n, are like pay dirt, containing dirt and apparent gold. They entice us to ask: which of these numbers are prime, which are not, and if not, why not - especially the shiny ones? We address the "why not?" of that question here, exploring the nature of all Mersenne composites, the other side of Mersenne ...
Klintberg, Amy
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On generalized Mersenne prime

SeMA Journal, 2014
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Hoque, Azizul, Saikia, Helen K.
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A Technique for Image Encryption Using the Modular Multiplicative Inverse Property of Mersenne Primes

Symmetry
Mersenne prime numbers, expressed in the form (2n − 1), have long captivated researchers due to their unique properties. The presented work aims to develop a symmetric cryptographic algorithm using a novel technique based on the logical properties ...
Shanooja M. A., Anil Kumar M. N.
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Mersenne Primes, Irrationality and Counting Subgroups

Bulletin of the London Mathematical Society, 1997
The author continues his studies on counting congruence subgroups in arithmetic subgroups [see ibid. 26, 255-262 (1994; Zbl 0849.11066)]. In this paper he considers the question of counting subgroups of \(p\)-power index in a group \(G_I\) which is the product of alternating groups \(A_{p^a}\) for \(a\in I\), where \(I\) is some subset of \(\mathbb{N}\)
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Testing Mersenne Primes with Elliptic Curves

2006
The current primality test in use for Mersenne primes continues to be the Lucas-Lehmer test, invented by Lucas in 1876 and proved by Lehmer in 1935. In this paper, a practical approach to an elliptic curve test of Gross for Mersenne primes, is discussed and analyzed.
Song Y. Yan, Glyn James
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Where is the next mersenne prime hiding?

The Mathematical Intelligencer, 1983
Almost identical to the paragraph 3.5 of the author's book reviewed above.
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Fast Mersenne prime testing on the GPU

Proceedings of the Fourth Workshop on General Purpose Processing on Graphics Processing Units, 2011
The Lucas-Lehmer test for Mersenne primality can be efficiently parallelized for GPU-based computation. The gpuLucas project implements an irrational-base discrete weighted transform approach (IBDWT) using balanced-integers, non-power-of-two transforms, and carry-save radix representations.
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