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): $1093$