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Let $p$ be a prime, and let $x\bmod p$ denote the least positive residue of $x$ modulo $p$. For $a \in \{1,2,...,p−1\}$, set $$f(a)=\sum^{p−1}_{g=1} g \times (ag \bmod p).$$

Is it true that $f(a)=f(b)$ if and only if $a=b$ or $a=b^{−1}$ mod $p$? Can we determine $f(a) \bmod p^2$?

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  • $\begingroup$ I am afraid that this question is not appropriate for this site - please read the FAQ. You might have better luck at math.stackexchange.com, but you should read the FAQ there also, in particular the comments about homework. $\endgroup$ Oct 14, 2011 at 12:44
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    $\begingroup$ Why was this closed? The notation is hard to read, but it seems he's asking about the residue mod $p^2$ (not mod $p$) of $\sum_{g=0}^{p-1} g [ag]$ where $[\cdot]$ is the least residue mod $p$, and that question looks nontrivial (though it may reduce to known identities for Dedekind sums). $\endgroup$ Oct 14, 2011 at 15:31
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    $\begingroup$ @Noam: Why do MO questions ever get closed? It's hard to read, not motivated at all, and appears to be homework that the OP is not even attempting to make palatable to MO users. Whether the mathematical question behind it is good or not becomes irrelevant at this point. $\endgroup$ Oct 14, 2011 at 16:11
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    $\begingroup$ Thanks to G.Myerson for the pointer to the math.se question. Since one cannot post a new answer to a closed question, I answered there and report here: numerically it seems $f(a)/p \equiv (a+a^{-1})/12 \bmod p$, which would imply the desired criterion for $f(a) \equiv f(b) \bmod p^2$. $\endgroup$ Oct 15, 2011 at 1:41
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    $\begingroup$ This question is clear now with the help of Noam D. Elkies and Gerry Myerson. See math.stackexchange.com/questions/72328/… $\endgroup$
    – fan
    Oct 15, 2011 at 7:30

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