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, 2013 With the boom in information technology and the penetration of these technologies in an increasing number of areas, such as electronic business, many questions about security arise. Data in electronic form bring many benefits, but they are vulnerable to various abuses. Therefore, data need to be adequately protected.
Lynn Margaret Batten
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Russian Mathematics, 2022
This article studies a combined primality test for natural numbers, called \textit{L2 test}, by combining the Lucas test and the Fermat condition test. The efficiency and complexity of this test are also analyzed, and a methodology for identifying composite numbers that pass the L2 test (called L2 pseudoprimes) is presented.
Ishmukhametov, S. T. +3 more
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This article studies a combined primality test for natural numbers, called \textit{L2 test}, by combining the Lucas test and the Fermat condition test. The efficiency and complexity of this test are also analyzed, and a methodology for identifying composite numbers that pass the L2 test (called L2 pseudoprimes) is presented.
Ishmukhametov, S. T. +3 more
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, 2008 Prime numbers have a very simple definition, but they inspire an abundance of interesting and complex theory. Originally a topic of interest only to pure mathematicians, primes have found an important practical role in our modern technological world ...
Jeff Wehrwein (14101011)
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ACM Communications in Computer Algebra, 2020
In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then ...
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In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then ...
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On a Modification of The Lucas Primality Test
Lobachevskii Journal of Mathematics, 2023Let \(F_n\) be the Fibonacci series defined by \(F_0 = 0\), \(F_1 = 1\), and \(F_{n+2} = F_n + F_{n+1}\). Set \(e(n) = (n/5)\), where \((\cdot/5)\) is the usual Legendre symbol. The classical Lucas primality test is based on the following result: If \(n > 5\), then \(F_{n-e(n)} \equiv 0 \pmod n\).
Ishmukhametov, Sh. +2 more
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Pseudocubes and Primality Testing
2004The recent ideas of Agrawal, Kayal, and Saxena have produced a milestone in the area of deterministic primality testing. Unfortunately, their method, as well as their successors are mainly of theoretical interest, as they are much too slow for practical applications.
Pedro Berrizbeitia +2 more
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Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92, 1992
Rabin’s algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm.
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Rabin’s algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm.
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A New Probabilistic Primality Test
Journal of Mathematical Sciences, 2020In this paper, a new efficient general probabilistic primality test is presented. The main idea is as follows. Let \(n > 1\) be an odd positive integer. First, it is checked whether \(n\) can be represented as \(n = a^b\), where \(a\) and \(b\) are integers \(\ge 2\).
Moshonkin, A. G., Khamitov, I. M.
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Primality testing with Lucas functions
1993A generalization of Fermat's Little Theorem is derived by using Lucas functions. This generalization yields new classes of pseudoprimes and can be used to improve some well-known primality tests.
Rudolf Lidl, Winfried B. Müller
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A primality test for Fermat numbers
1995The paper gives a primality criterion for Fermat numbers \(F_n=2^{2^n}+1\) \(\left(n=0,1,2,\ldots\right)\). The author proves the following theorem. Let \(k\) and \(n\) be fixed positive integers such that \(0< k\leq [\log n/\log_2]\) and \(n>1\). The Fermat number \(F_k\) is prime if and only if \(F_k\) does not divide \(T\left(2^{n-1}\right)\), where
Grytczuk, A., Grytczuk, J.
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