Wieferich's Criterion

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

  • Definition:Wieferich Prime


Source of Name

This entry was named for Arthur Josef Alwin Wieferich.


Historical Note

Arthur Wieferich discovered what is now known as 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): $1093$
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