Solutions to p^2 Divides 10^p - 10
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Theorem
The known prime numbers $p$ which satisfy the equation:
- $p^2 \divides \paren {10^p - 10}$
where $\divides$ denotes divisibility, are:
- $3, 487, 56 \, 598 \, 313$
This sequence is A045616 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
This theorem requires a proof. In particular: Some sort of computer program can be implemented, I suppose You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also see
Sources
- July 1993: Peter L. Montgomery: New Solutions of $a^{pā1} \equiv 1 \pmod {p^2}$ (Math. Comp. Vol. 61, no. 203: pp. 361 ā 363) www.jstor.org/stable/2152960
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $487$