# Solutions to p^2 Divides 10^p - 10

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## Contents

## 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

## 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): $487$