On the factors of certain Mersenne numbers. II
نویسندگان
چکیده
منابع مشابه
Generalised 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....
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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 ...
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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.
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We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers Mp in the range 17000 < p < 105.
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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: Mathematics of Computation
سال: 1964
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1964-0159776-x