<sod>* Let ''n'' be an odd [[square-free]] integer. Assuming the Birch and Swinnerton-Dyer conjecture, ''n'' is the area of a right triangle with rational side lengths (a [[congruent number]]) if and only if the number of triplets of integers (''x'', ''y'', ''z'') satisfying <math>2x^2 + y^2 + 8z^2 = n</math> is twice the number of triples satisfying <math>2x^2 + y^2 + 32z^2 = n</math>. This statement, due to [[Tunnell's theorem]] {{harv|Tunnell|1983}}, is related to the fact that ''n'' is a congruent number if and only if the elliptic curve <math>y^2 = x^3 - n^2x</math> has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its ''L''-function has a zero at 1). The interest in this statement is that the condition is easily verified.<ref>{{Cite book<eod>
<sod>* Let {{mvar|n}} be an odd [[square-free]] integer. Assuming the Birch and Swinnerton-Dyer conjecture, {{mvar|n}} is the area of a right triangle with rational side lengths (a [[congruent number]]) if and only if the number of triplets of integers ({{mvar|x}}, {{mvar|y}}, {{mvar|z}}) satisfying {{math|2''x''<sup>2</sup> + ''y''<sup>2</sup> + 8''z''<sup>2</sup> {{=}} ''n''}} is twice the number of triples satisfying {{math|2''x''<sup>2</sup> + ''y''<sup>2</sup> + 32''z''<sup>2</sup> {{=}} ''n''}}. This statement, due to [[Tunnell's theorem]] {{harv|Tunnell|1983}}, is related to the fact that ''n'' is a congruent number if and only if the elliptic curve {{math|''y''<sup>2</sup> {{=}} ''x''<sup>3</sup> − ''n''<sup>2</sup>''x''}} has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its {{mvar|L}}-function has a zero at {{math|1}}). The interest in this statement is that the condition is easily verified.<ref>{{Cite book<eod>