Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 639.10014
Autor: Erdös, Paul; Stewart, C.L.; Tijdeman, R.
Title: Some diophantine equations with many solutions. (In English)
Source: Compos. Math. 66, No.1, 37-56 (1988).
Review: The authors of this interesting paper consider three types of diophantine problems, having common roots of proofs. First they prove that there exist distinct positive integers a1,...,ak,b1,...,b\ell such that the greatest prime factor of prodki = 1prod\ellj = 1(ai+bj) is small, so that the lower estimate of K. Györy, C. L. Stewart and R. Tijdeman [Compos. Math. 59, 81-88 (1986; Zbl 602.10031)] is not far from being best possible. Secondly, the number of coprime solutions of S-unit equations x+y = z is examined, where x,y,z are composed of a fixed finite set of primes. Thirdly, the number of solutions of Thue-Mahler equations is studied. It is shown, that the last two problems may have surprisingly many solutions in contrast with the upper bounds of J. H. Evertse [Invent. Math. 75, 561-584 (1984; Zbl 521.10015)].
Reviewer: I.Gaál
Classif.: * 11D41 Higher degree diophantine equations
11D57 Multiplicative and norm form diophantine equations
11D75 Diophantine inequalities
Keywords: diophantine inequalities; sums of integers; greatest prime factor; number of coprime solutions; S-unit equations; Thue-Mahler equations
Citations: Zbl 602.10031; Zbl 521.10015
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