منابع مشابه
Mersenne and Fermat Numbers
The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...
متن کاملGeneralised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked....
متن کاملOverpseudoprimes, Mersenne Numbers and Wieferich Primes
We introduce a new class of pseudoprimes-so called ”overpseudoprimes” which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2 − 1, where p is prime, and squa...
متن کاملOn the largest prime factor of the Mersenne numbers
Let P (k) be the largest prime factor of the positive integer k. In this paper, we prove that the series ∑ n≥1 (log n)α P (2n − 1) is convergent for each constant α < 1/2, which gives a more precise form of a result of C. L. Stewart of 1977.
متن کاملSums of Prime Divisors and Mersenne Numbers
The study of the function β(n) originated in the paper of Nelson, Penney, and Pomerance [7], where the question was raised as to whether the set of Ruth-Aaron numbers (i.e., natural numbers n for which β(n) = β(n+ 1)) has zero density in the set of all positive integers. This question was answered in the affirmative by Erdős and Pomerance [5], and the main result of [5] was later improved by Po...
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ژورنال
عنوان ژورنال: SSRN Electronic Journal
سال: 2018
ISSN: 1556-5068
DOI: 10.2139/ssrn.3183800