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A remark on prime divisors of lengths of sides of Heron triangles. (English) Zbl 1096.11011

A Heron triangle is a triangle whose lengths of all three sides as well as its area are positive integers. Fix a finite set \(P\) of primes and denote by \(S\) the set of integers divisible only by primes of \(P\). In this paper it is proved that there are finitely many Heron triangles whose lengths of sides, say \(a,b,c\), are relatively prime and belong to \(S\). Moreover, if \(P\) contains only one prime \(\equiv 1\pmod{4}\), then these triangles can be effectively determined. For \(P=\{2,3,5,7,11\}\) the authors make one further brave step and determine explicitly all these triangles, 12 in number, as it turns out, the greatest one being that with sides \((625,625,672)\). It seems that, with our nowadays computers, no one can be more brave to deal with a set \(P\) containing more than 5 primes. Basic tools for the proofs are, a result of Evertse et al. on \(S\)-unit equations [J.-H. Evertse, K. Gőry, C. L. Stewart and R. Tijdeman, \(S\)-unit equations and their applications, New Advances in Transcendence Theory, Durham 1986, Cambridge University Press, 110–174 (1988; Zbl 0658.10023)], two results from B. M. M. de Weger’s thesis [Algorithms for Diophantine equations, CWI Tracts, 65. Centrum voor Wiskunde en Informatica, Amsterdam (1989; Zbl 0687.10013)] and a result of K. Yu [“\(p\)-adic logarithmic forms and group varieties. II”, Acta Arith. 89, 337–378 (1999; Zbl 0928.11031)].

MSC:

11D25 Cubic and quartic Diophantine equations
11D57 Multiplicative and norm form equations
11Y50 Computer solution of Diophantine equations
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