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ddxexex wrote:

This isn't a super-formal proof but it gets the point across

the expression is (1+n)^p - n^p - 1. The translation is just plain wrong with the formula.

this is equal to (1+n)^p - (n^p+1)

since p is prime it must be >= 2. So we can always expand this to at least three terms.

expand out the first part out

**1** + p*(1+n)+ ... + p*(1+n)^(p-1)+

**n^p** -

**(1+n^p)**
rearrange

*(1+n)+ ... + p*(1+n)^(p-1) +

**(1 +n^p) - (1+n^p)**
leaving

p*(1+n)+ ... + p*(1+n)^(p-1)

since all terms have a p in it it is divisible by p

The binomial expansion proof is correct. Well done.

The other accepted proof is using fermat's little theorem which someone mentioned before as well.

But before you get smug, This question is taken out of Mahou Shojo Madoka Magica, where the characters are in middle school. I donno about where other people live but in the UK, binomial theorem is definitely beyond and above middle school syllabus...