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On the distribution of \(\tau\)-congruent numbers. (English) Zbl 1346.14085

Congruent numbers are positive integers which are the area of a right triangle with rational sides. The integer \(n\) is a congruent number if and only if the elliptic curve \(E^{(n)} : y^2=x(x^2-n^2)\) has a rational point of order different from 2. If, for some nonzero rational number \(\tau\), the curve \(E^{(n)}_\tau : y^2=x(x-n\tau)(x-n\tau^{-1})\) enjoys the same property, then \(n\) is called \(\tau\)-congruent number (and is the area of a triangle with rational sides). The authors show that the formula \(n=cdrs(cr+ds)(dr-cs)\) provides a \(\tau=\frac{c}{d}\)-congruent number \(n>0\) for almost all coprime integers \(r\) and \(s\) and use this to show that, for any integer \(m>1\) and any nonzero rational \(\tau\), there exists infinitely many \(\tau\)-congruent numbers in each residue class modulo \(m\).

MSC:

14H52 Elliptic curves
11G05 Elliptic curves over global fields
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References:

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