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On prime factors of integers of the form (ab+1)(bc+1)(ca+1)

K. GyőryA. Sárközy — 1997

Acta Arithmetica

1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c)...

On partitions without small parts

J.-L. NicolasA. Sárközy — 2000

Journal de théorie des nombres de Bordeaux

Let r ( n , m ) denote the number of partitions of n into parts, each of which is at least m . By applying the saddle point method to the generating series, an asymptotic estimate is given for r ( n , m ) , which holds for n , and 1 m c 1 n log n c 2 .

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