Editing Bernoulli number (section)

=== The Kummer theorems ===
The Bernoulli numbers are related to [[Fermat's Last Theorem]] (FLT) by [[Ernst Kummer|Kummer]]'s theorem,{{r|Kummer1850}} which says:

:If the odd prime {{mvar|p}} does not divide any of the numerators of the Bernoulli numbers {{math|''B''<sub>2</sub>, ''B''<sub>4</sub>, ..., ''B''<sub>''p'' − 3</sub>}} then {{math|''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> {{=}} 0}} has no solutions in nonzero integers.

Prime numbers with this property are called [[regular prime]]s. Another classical result of Kummer are the following [[Modular arithmetic#Congruence|congruences]].{{r|Kummer1851}}

{{main|Kummer's congruence}}

:Let {{mvar|p}} be an odd prime and {{mvar|b}} an even number such that {{math|''p''&nbsp;−&nbsp;1}} does not divide {{mvar|b}}. Then for any non-negative integer {{mvar|k}}
:: <math> \frac{B_{k(p-1)+b}}{k(p-1)+b} \equiv \frac{B_{b}}{b} \pmod{p}. </math>

A generalization of these congruences goes by the name of {{math|''p''}}-adic continuity.