Rank of the elliptic curve $y^2=x^3+px$
Solution 1:
This is shown in Silverman's "The Arithmetic of Elliptic Curves", Chapter X, Section 6 (The curve $Y^2=X^3+DX$), Proposition 6.2.
Solution 2:
For $2M^4-2pe^4=N^2$, we see that $N=2n$ for some integer $n$. Substituting in the equation and reducing by 2 we get $2n^2=M^4-pe^4$. Now if $p\equiv 7(mod 16)$ then we get $n^2\equiv -3 (mod 16)$ so $n^2\equiv -3 (mod 8)$ which is a contradiction. Because $x^2\equiv 0,1,4(mod 8) $ for integer $x$.