$\frac{a^{239}+1}{a}=\frac{b^{239}+1}{b}$ $ab( a^{238}-b^{238})-(a-b)=0$ $ab( a^{237}+a^{236}b+...+b^{237})=1$ $1=ab(a^{237}+...+b^{237}) > ab * 238 \sqrt[238]{(ab)^{\frac{237*238}{2}}}=238*(ab)^{\frac{239}{2}}$ So $238^2 (ab)^{239} < 1$

We can see $a^{239}-b^{239}=c(a-b)$ . By divide $a-b$ we can see : $$a^{238}+a^{237}b+...+b^{238} = c$$( ) If we put this reasualt in main equation we get $c=1$ . ( ) By ( ) and ( ) we get : $$1 = a^{238}+a^{237}b+...+b^{238} > 238(ab)^{\frac{239}{2}}$$ And inequality is strict beacuse $a \neq b $ .$\blacksquare$