Conjecture of Agoh-Giuga

In Théorie of the numbers, the conjecture of Agoh-Giuga about the numbers of Bernoulli $B\_k\; \backslash ,$ postulates that p is a Prime number if and only if

$pB\_\; \{p-1\}\; \backslash \; equiv\; -1\; \backslash \; MOD\; \{p\}\; \backslash ,$.

The conjecture thus stated is due to Takashi Agoh; an equivalent formulation is due to Giuseppe Giuga, starting from 1950, with the effect of the primality of p if and only if

$1^\; \{p-1\}\; +2^\; \{p-1\}\; +\; \backslash \; cdots\; +\; (p-1)\; ^\; \{p-1\}\; \backslash \; equiv\; -1\; \backslash \; pmod\; p\; \backslash ,$.

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