Let

1 + 1/2 + 1/3 +...+1/823 = r/s*823.

Without calculating the LHS, prove that r= s (mod 823^3).

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- May 9th 2009, 10:09 PMCairoWolstenholme
Let

1 + 1/2 + 1/3 +...+1/823 = r/s*823.

Without calculating the LHS, prove that r= s (mod 823^3). - May 11th 2009, 09:12 AMCairo
Put another way,

How do I show that p^3 divides r-s in this instance? - May 11th 2009, 09:30 AMMedia_ManWolstenholme's Theorem
Well, if , then . Since is prime, by Wolstenholme's Theorem, the numerator of is divisible by , and by definition, is the lowest positive integer ensuring the RHS is a whole number, essentially doing the job of canceling all the denominators: , for some and .

Therefore, the entire RHS is divisible by , hence

*Are you asking us to prove Wolstenholme's Theorem or to simply connect the dots enough to make use of this theorem?