THEOREM OF THE DAY: Tunnell’s Theorem

Let $n$ be a square-free positive integer and denote by $S_n(a, b, c)$ the number of solutions in integers, $x, y,z$, of the equation 

$ax^2 + by^2 + cz^2 = n$.

Click on the image.

Then a necessary condition for $n$ to be a congruent number is that 

$S_n(2, 1, 8) = 2S_n(2, 1, 32)$ $n$ odd 

and 

$S_n(8, 2, 16) = 2S_n(8, 2, 64)$ $n$ even. 

Moreover, if the Birch and Swinnerton-Dyer conjecture is true then this condition is also sufficient