Congruent numbers, quadratic forms and $$K_2$$

We give a new and self-contained proof for an alternative form of Tunnell’s theorem on congruent numbers. The proof relies on more knowledge on quadratic forms with less calculation than Tunnell’s proof. The approach is based on our observation that the equality $$\#\{(x,y,z,w) \in \mathbb {Z}^4|p=2x^2+3y^2+3z^2+4w^2+2yz\}=\#\{(x,y,z) \in \mathbb {Z}^3|p^2=2x^2+4y^2+9z^2-4yz\}$$ holds for any odd prime p. The same method is applied to the elliptic curves $$E_n:~y^2=x^3+n^2x$$ and $$E_{n}:~~y^2=x(x-n)(x+3n)(E_{-n}:~~y^2=x(x+n)(x-3n))$$ ( $$\pi /3$$ ( $$2\pi /3$$ )-congruent elliptic curves), where we list, without proof, the analogous results. A series of criteria for congruent numbers are given. In particular, for a prime p, we show that if $$p\equiv 1\pmod {8}$$ is a congruent number then the 8-rank of $$K_2O_{\mathbb {Q}(\sqrt{p})}$$ equals one;  if $$p\equiv 1\pmod {16}$$ with $$h(-p)\not \equiv h(-2p)\pmod {16}$$ then 2p is not a congruent number; and if $$p\equiv 1, q\equiv 3\pmod {8}$$ are two primes with $$h(-pq)\not \equiv h(-p)\pmod {8}$$ then pq is not a congruent number.