TITLE: Simple Method of Solution of $X^p+Y^p=(X+1)^p$
QUESTION [9 upvotes]: I am looking for simple prove of non existence of solution of $$X^p+Y^p=(X+1)^p$$
for $p>2$
I know it is partial case of Fermat Last Theorem. But I am looking for simple method without going into the complex part of group theory and algebra.
I have simply found from the Fermat Little Theorem that $$Y\equiv1\pmod p$$
But this the maximum I was able to get. Is there any idea how to solve this using simple methods?

REPLY [3 votes]: This is an open question. See Fermat’s Last Theorem for Amateurs by Paulo Ribenboim. Specifically following results of Catalan are shown: 

For $p$ odd prime number, $0<x<y$ integers such that $x^p+y^p=(y+1)^p$, the following is true:

$py(y+1)$ divides $x^p-1$
$p\nmid x$,  $p\mid x-1$
If $q$ is a prime dividing $y+1-x$ then $q$ divides $x-1$
$\gcd(x+y,y+1-x)=1$
$\gcd(2x-1,2y+1)=1$
$x$ is the only integer such that $(py^{p-1})^{1/p} < x < (p(y+1)^{p-1})^{1/p}$