The impossibility of a nonzero integer solution $r^4-s^4=v^2$ can also be shown

by a more direct descent that avoids some of the steps used by Fermat. The main

steps are as follows, assuming $r,s,$ and hence $v$ have no common prime divisor.

$r^4-s^4=v^2$Longrightarrow,$r^2=a^2+b^2,s^2=2ab,v=a^2-b^2$

for some nonzero integers $a,b$

$=>,a=c^2-d^2,b=2cd$

for some nonzero integers $c,d$

Longrightarrow,$c=e^2,d=f^2$ and $c^2-d^2$ are squares

because $s^2=4cd(c^2-d^2)$

and $c,d,c^2-d^2$ have no common prime divisor

Longrightarrow,$e^4-f^4=g^2$

for an integer pair $(e,f)$ smaller than $(r,s).$

$11\mathrm{.}4\mathrm{.}4$ Justify the steps in this argument.