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On the distribution of congruent numbers. (English) Zbl 1228.11083

A positive integer is said to be a congruent number if it is equal to the area of a right triangle with rational sides. It is known that a positive integer \(n\) is a congruent number if and only if the elliptic curve \(y^2=x(x^2-n^2)\) has positive Mordell-Weil rank. In this article the authors prove that for any integer \(m>1\), any congruence class modulo \(m\) contains infinitely many congruent numbers \(n\), inequivalent modulo squares, such that the rank of the elliptic curve \(y^2=x(x^2-n^2)\) is greater than or equal to two.

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations

Software:

Magma
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References:

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