TITLE: Is there a elementary proof that $3$ is not a congruent number?
QUESTION [7 upvotes]: How to show  that $3$ is not a congruent number? 
I don't want to use Tunnell's theorem(1982).
Is there a more elementary proof？Thanks a lot!
There is proof in Judith D. Sally's book "Roots to Research", but I don't have a copy...

REPLY [4 votes]: It is elementary to show that $n$ is not congruent iff the elliptic curve $E_n: y^2=x^3-n^2x$ has no non-trivial rational point. For $n=3$ it is not too difficult to show (I hope) that $E_3$ has rank zero (due to Genocci $1855$, without Tunnels result). There is a complete $2$-descent for the curves $E_p$ with $p$ an odd prime, which shows that the rank is zero if
$p\equiv 3 \mod 8$, see http://math.stanford.edu/~rhoades/FILES/EC.pdf, section $26.1$.