Divisibility problem

**gabor7896** · January 28th 2010, 11:16 AM

Let a+b=p, where a and b are positive integer numbers and p is a prime. Prove that $a^p+b^p$ is divisible by $p^2$ , but it is not divisible by $p^3$ !

**Drexel28** · January 28th 2010, 02:42 PM

Originally Posted by gabor7896

Let a+b=p, where a and b are positive integer numbers and p is a prime. Prove that $a^p+b^p$ is divisible by $p^2$ , but it is not divisible by $p^3$ !

$a^p+b^p=\left(a+b\right)^p-\sum_{k=1}^{p-1}{p \choose k}a^{p-k}b^k$ , start with that.

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