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Thread: Wolstenholme

  1. #1
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    Wolstenholme

    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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  2. #2
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    Put another way,

    How do I show that p^3 divides r-s in this instance?
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  3. #3
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    Atlanta, GA
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    Re: Wolstenholme

    Well, if $r=823s\sum_{n=1}^{823}\frac{1}{n}$, then $r-s=823s\sum_{n=1}^{822}\frac{1}{n}$. Since $823$ is prime, by Wolstenholme's Theorem, the numerator of $\sum_{n=1}^{822}\frac{1}{n}$ is divisible by $823^2$ , and by definition, $s$ is the lowest positive integer ensuring the RHS is a whole number, essentially doing the job of canceling all the denominators: $r-s=823s\sum_{n=1}^{822}\frac{1}{n}=823s\frac{N}{D}= 823s\frac{823^2k}{D}=823^3\frac{sk}{D}=823^3s'k$, for some $s'$ and $k$.

    Therefore, the entire RHS is divisible by $823^3$, hence $r \equiv s (\bmod 823^3)$

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