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On sums and products of residues modulo p

100%

Sárközy A.

Acta Arithmetica

2005

tom 118

nr 4

403-409

On isolated, respectively consecutive large values of arithmetic functions

100%

Sárközy A.

Acta Arithmetica

1994

tom 66

nr 3

269-295

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

63%

Győry K. , Sárközy A.

Acta Arithmetica

1997

tom 79

nr 2

163-171

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 𝓐 is a finite set of triples (a,b,c) of positive integers a, b, c with the property mentioned above then for some (a,b,c) ∈ 𝓐, (1) is greater than a constant times log|𝓐|loglog|𝓐|, where |𝓐| denotes the cardinality of 𝓐 (cf. Corollary to Theorem 1). Further, we show that this bound cannot be replaced by $|𝓐|^ε$ (cf. Theorem 2). Recently, Stewart and Tijdeman [9] proved the conjecture in another special case. Namely, they showed that if a ≥ b > c then (1) exceeds a constant times log((loga)/log(c+1)). In the present paper we give an estimate from the opposite side in terms of a (cf. Theorem 3).

Some solved and unsolved problems in combinatorial number theory, ii

63%

Erdős P. , Sárközy A.

Colloquium Mathematicum

1993

tom 65

nr 2

201-211

In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.

On the number of prime factors of integers of the form ab + 1

51%

Győry K. , Sárközy A. , Stewart C.

Acta Arithmetica

1996

tom 74

nr 4

365-385

On pseudorandom binary lattices

51%

Hubert P. , Mauduit C. , Sárközy A.

Acta Arithmetica

2006

tom 125

nr 1

51-62

On the number of pairs of partitions of n without common subsums

51%

Erdős P. , Nicolas J.-L. , Sárközy A.

Colloquium Mathematicum

1992

tom 63

nr 1

61-83

On the divisibility properties of sequences of integers (II)

38%

Erdös P. , Sárközy A. , Szemerédi E.

Acta Arithmetica

1968

tom 14

nr 1

1-12

On the divisibility properties of integers (I)

32%

Erdös P. , Sárközy A. , Szemerédi E.

Acta Arithmetica

1965-1966

tom 11

nr 4

411-418

Some asymptotic formulas on generalized divisor functions, III

32%

Sárközy A. , Erdös P.

Acta Arithmetica

1982

tom 41

nr 4

395-411

Über ein Problem von Erdös und Moser

26%

Sárközy A. , Szemerédi E.

Acta Arithmetica

1965-1966

tom 11

nr 2

205-208

On sums of sequences of integers, I

26%

Balog A. , Sárközy A.

Acta Arithmetica

1984-1985

tom 44

nr 1

73-86

On a theorem of Erdös and Fuchs

26%

Sárközy A.

Acta Arithmetica

1980

tom 37

nr 1

333-338

Über totalprimitive Folgen

23%

Sárközy A.

Acta Arithmetica

1962-1963

tom 8

nr 1

21-31

Über reduzible Folgen

20%

Sárközy A.

Acta Arithmetica

1964-1965

tom 10

nr 4

399-408