Results 1 to 10 of about 457 (186)
Features of digital signal processing algorithms using Galois fields GF(2n+1). [PDF]
PLoS ONE, 2023 An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1 ...
Ibragim E Suleimenov +2 more
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Gaussian Mersenne and Eisenstein Mersenne primes [PDF]
Mathematics of Computation, 2010 Summary: The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas-Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth's Theorem.
Berrizbeitia, Pedro, Iskra, Boris
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The 24th mersenne prime. [PDF]
Proc Natl Acad Sci U S A, 1971 The 24th Mersenne prime M p = 2 p - 1, and currently the largest known prime, is 2 19937 - 1. Primality was shown by the Lucas-Lehmer test on an IBM 360/91 computer.
Tuckerman B.
europepmc +4 more sources
The 25th and 26th Mersenne Primes [PDF]
Mathematics of Computation, 1980 The 25th and 26th Mersenne primes are 2 21701 − 1 {2^{21701}} - 1 and 2 23209 − 1 {2^{23209}} - 1 , respectively.
Noll, Curt, Nickel, Laura
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BiEntropy, TriEntropy and Primality [PDF]
Entropy, 2020 The order and disorder of binary representations of the natural numbers < 28 is measured using the BiEntropy function. Significant differences are detected between the primes and the non-primes.
Grenville J. Croll
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Mathematics of Computation, 1991 The number 2 110503 − 1 {2^{110503}} - 1 is a Mersenne prime. There are exactly two Mersenne exponents between 100000 and 139268, and there are no Mersenne exponents between 216092 and 353620. Thus, the number
W. N. Colquitt, L. Welsh
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The specifics of the Galois field GF(257) and its use for digital signal processing [PDF]
Scientific ReportsAn algorithm of digital logarithm calculation for the Galois field $$GF(257)$$ G F ( 257 ) is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 ...
Akhat Bakirov +4 more
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Mersenne Primes in Real Quadratic Fields [PDF]
, 2012 The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenne primes in the form $x^{2}+7y^{2}$. It is also proved that $x$ is divisible by 8 and $y\equiv \pm3\pmod{8}$ generalizing the result of F Lemmermeyer ...
Palimar, Sushma, Shankar, B. R.
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Efficient arithmetic in (pseudo-)mersenne prime order fields
Advances in Mathematics of Communications, 2022 <p style='text-indent:20px;'>Elliptic curve cryptography is based upon elliptic curves defined over finite fields. Operations over such elliptic curves require arithmetic over the underlying field. In particular, fast implementations of multiplication and squaring over the finite field are required for performing efficient elliptic curve ...
Nath, Kaushik, Sarkar, Palash
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Congruence Properties of Mersenne Primes [PDF]
, 2011 In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer 'n'} and generalize the result to all powers of 2. We also tabulate and show their relationship with other whole numbers up to 10.
Srinath, M. S. +2 more
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