On the greatest prime factor of (ab + 1)(bc + 1)(ca + 1)
نویسندگان
چکیده
منابع مشابه
ON THE GREATEST PRIME FACTOR OF (ab+ 1)(ac+ 1)
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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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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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.
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Let P (m) denote the greatest prime factor of m. For integer a > 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (a − 1) ,a n2− for any > 0. We study analogues results for the corresponding divisibility sequence over the function field Fq(t) and for some divisibility sequences associated to elliptic curves over the rational field Q. In honor of M. Ram Murt...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1998
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-86-1-45-49