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

K. Győry, A. 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 rough and smooth neighbors.

William D. Banks, Florian Luca, Igor E. Shparlinski (2007)

Revista Matemática Complutense

We study the behavior of the arithmetic functions defined byF(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1)where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.

On the distribution of consecutive square-free primitive roots modulo p

Huaning Liu, Hui Dong (2015)

Czechoslovak Mathematical Journal

A positive integer n is called a square-free number if it is not divisible by a perfect square except 1 . Let p be an odd prime. For n with ( n , p ) = 1 , the smallest positive integer f such that n f ≡ 1 ( mod p ) is called the exponent of n modulo p . If the exponent of n modulo p is p - 1 , then n is called a primitive root mod p . Let A ( n ) be the characteristic function of the square-free primitive roots modulo p . In this paper we study the distribution ∑ n ≤ x A ( n ) A ( n + 1 ) , and give an asymptotic formula by using properties of character sums.

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