# Wieferich's Criterion

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## Theorem

Suppose Fermat's equation:

- $x^p + y^p = z^p$

has a solution in which $p$ is an odd prime that does not divide any of $x$, $y$ or $z$.

Then $2^{p - 1} - 1$ is divisible by $p^2$.

## Proof

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## Also known as

Some sources give this as **Wieferich's theorem**, but this is also used for his result concerning the Hilbert-Waring Theorem for cubes.

## Also see

## Source of Name

This entry was named for Arthur Josef Alwin Wieferich.

## Historical Note

Arthur Wieferich discovered what is now known as Wieferich's Criterion in $1909$.

It had profound implications for Fermat's Last Theorem, in that it demonstrated that the only cases that needed to be considered were those for the Wieferich primes, of which only $2$ are known less than $10^{17}$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1093$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1093$