## 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

### Keywords:

congruent numbers; elliptic curves
Full Text:

### References:

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