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Recall that Giuga's conjecture (1950), still widely open, asserts: let $n$ be a positive integer, if $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime.

In light of the question and answers showing Gauss' generalization of Wilson's theorem is implied by group-theoretical results (classification of $n$ for which the units mod $n$ are cyclic), I'd like to ask:

are there group-theoretical results that make it difficult to believe that Giuga's conjecture is true?

Partial answers or numerical considerations welcome.

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    $\begingroup$ I doubt it. this is a sum of powers which means we need a ring structure rather than just a group. And why do you speculate that it is hard to believe that the conjecture is true? $\endgroup$ Commented Nov 12 at 9:26
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    $\begingroup$ Thank you for your reply. Maybe I didn't phrase my question well. I'm not speculating that the conjecture is hard to believe (the numerics so far says otherwise) I was just wondering if, when looked at from a group theory angle, i.e. a relationship between cardinals of some groups, some known results or conjectures would be in tension with it. $\endgroup$ Commented Nov 12 at 10:22
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    $\begingroup$ Maybe you have already looked into the polynomial $P_n(t) = \prod_{k=1}^{n-1}(t-k^{n-1})$. Using Fermats and Wilsons theorem one can prov:e $n$ is prime $\iff$ $P_n(t) \equiv \sum_{k=0}^{n-1} t^k \mod(n)$. Giuga's conjecture and Wilson's theorem concern special coefficients of this polynomial. $\endgroup$ Commented Nov 12 at 11:40
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    $\begingroup$ @mathoverflowUser yes, thank you, I had but never got far. There must be an invariant somewhere. $\endgroup$ Commented Nov 13 at 6:48

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