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$ a \cdot (a-1) \ \mid \ n^{a}-n \ \ \ \forall \ n \in \mathbb{N} $

Find all $a$.

$$$$

Obviously $a$ has to be prime.

I found $a=2,3,7$ and $43$ as solutions.

Note that these 4 solutions are the first elements of Sylvester's Sequence. $$$$

Are there more solutions?

Martin Hopf
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    Apply Korselt's criterion for Carmichael numbers. – Bill Dubuque Jun 09 '19 at 18:30
  • Korselt's criterion implies $a-1$ to be squarefree. For this the next squarefree canidate for $a-1$ respectively $a$ in the sequence is: $2 \cdot 3 \cdot 7 \cdot 43 + 1 = 1807 = 13 \cdot 139$. It is not a prime! I srongly tend to say that this breaks the chain. So the above 4 solutions are the only ones. – Martin Hopf Jun 10 '19 at 16:27
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    I checked the next two numbers (5th and 6th) in the sequence and they both don't work. The sixth one is prime, but that doesn't seem to matter. I also suspect these are the only solutions, but I haven't been able to prove it yet. – QC_QAOA Jun 10 '19 at 16:54

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