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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References:

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