On the greatest prime factor of $p-1$ with effective constants
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چکیده
منابع مشابه
On the greatest prime factor of p-1 with effective constants
Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p−1 exceeding (p−1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x > x0 there is a p < x with the greatest prime factor of p − 1 exceeding x 3 5 . The novelty of our approach is the avoid...
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We prove that for integers a > b > c > 0, the greatest prime factor of (ab+1)(ac+1) tends to infinity with a. In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product (ab + 1)(ac + 1)(bc + 1). In the paper [GSS], Gÿory, Sarkozy and Stewart conjectured that, for positive integers a > b > c, the greatest prime factor of the prod...
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We prove that whenever A and B are dense enough subsets of {1, . . . , N}, there exist a ∈ A and b ∈ B such that the greatest prime factor of ab + 1 is at least N1+|A|/(9N).
متن کاملOn the greatest prime factor of integers
Let N be a positive integer and let A and B be dense subsets of {1, . . . , N}. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab+ 1 as a and b run over the elements of A and B respectively. 1991 AMS Mathematics Subject Classification. Primary 11N30, Secondary 11L05, keywords: greatest prime factor, Selberg’s sieve, Kloosterman sums.
متن کاملOn the greatest prime factor of (ab+ 1)(ac+ 1)(bc+ 1)
Recently, Györy [2] has proved that (1) holds provided that at least one of P (a), P (b), P (c), P (a/b), P (a/c) and P (b/c) is bounded. While we have not been able to prove (1) we have been able to prove that if, a, b, c and d are positive integers with a 6= d and b 6= c then P ((ab+ 1)(ac+ 1)(bd+ 1)(cd+ 1))→∞ as the maximum of a, b, c and d tends to infinity. Notice, by symmetry, that there ...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2005
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-05-01749-7