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James
Lv 4
James asked in Science & MathematicsMathematics · 3 years ago

Is there a proof which shows that for all primes p, n^p-n is divisible by p?

I've shown that for all positive integers n that n^5-5 is divisible by 5.

I've also found that this seems to be the case for p|(n^p-n), given that p is one of the first five primes. And that it is not true that p divides n^p-n when p is a composite positive integer.

Update:

Excuse me,

I've shown that for all positive integers n that n^5-n is divisible by 5. Typo above.

2 Answers

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  • Robert
    Lv 7
    3 years ago
    Favorite Answer

    Factor n^p - n into n(n^[ p-1 ] - 1).

    Then, according to Fermat's "little" theorem,

    p | n^(p-1) - 1

  • DWRead
    Lv 7
    3 years ago

    n^5-5 is not divisible by 5 for n = 1.

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