Results 1 to 10 of about 14,157 (261)
Entropy, Periodicity and the Probability of Primality [PDF]
EntropyThe distribution of prime numbers has long been viewed as a balance between order and randomness. In this work, we investigate the relationship between entropy, periodicity, and primality through the computational framework of the binary derivative.
Grenville J. Croll
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Machine learning of the prime distribution. [PDF]
PLoS OneIn the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy–Ramanujan Theorem. We also provide a theoretical argument explaining the experimental observations of Y.–H. He about the learnability of primes, and posit that the Erdős–Kac law would very unlikely be discovered
Kolpakov A, Rocke AA.
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A primality test for Kpⁿ⁺¹ numbers and a generalization of Safe primes and Sophie Germain primes [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 In this paper, we provide a generalization of Proth's theorem for integers of the form Kpⁿ⁺¹. In particular, a primality test that requires a modular exponentiation (with a proper base a) similar to that of Fermat's test without the computation of any ...
Abdelrahman Ramzy
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On the calculation of integer sequences, associated with twin primes
Lietuvos Matematikos Rinkinys, 2023 The twin primes conjecture states that there are infinitely many twin primes. While studying this hypothesis, many important results were obtained, but the problem remains unsolved.
Igoris Belovas +2 more
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A Remark on the Factorization of Factorials [PDF]
Mathematics Interdisciplinary Research, 2021 The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. We show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all
Mehdi Hassani, Mahmoud Marie
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The Prime state and its quantum relatives [PDF]
Quantum, 2020 The Prime state of $n$ qubits, ${|\mathbb{P}_n{\rangle}}$, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than $2^n$. This state encodes, quantum mechanically, arithmetic properties of
D. García-Martín +4 more
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Coordinate distribution of Gaussian primes
Journal of the European Mathematical Society, 2021 We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.
Friedlander, John, Iwaniec, Henryk
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Arab Journal of Basic and Applied Sciences, 2023 Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy.
Moustafa Ibrahim
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THE DISTRIBUTION OF PRIME NUMBERS [PDF]
Nature, 2007 This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.
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An Analytic Approximation to the Density of Twin Primes
Recoletos Multidisciplinary Research Journal, 2018 The highly irregular and rough fluctuations of the twin primes below or equal to a positive integer x are considered in this study. The occurrence of a twin prime on an interval [0,x] is assumed to be random.
Dionisel Y. Regalado, Rodel Azura
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