Agoh–Giuga conjecture

In number theory the Agoh–Giuga conjecture on the Bernoulli numbers "B"_"k" postulates that "p" is a prime number if and only if

:$pB\_\{p-1\}\; equiv\; -1\; pmod\; p.$

The conjecture as stated is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that "p" is prime iff

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

References

*Agoh, T, "On Giuga’s conjecture" "Manuscripta Math.", 87(4), 501–510 (1995).
*Borwein, D.; Borwein, J. M., Borwein, P. B., and Girgensohn, R. "Giuga's Conjecture on Primality", "American Mathematical Monthly", 103, 40–50, (1996). [http://www.math.uwo.ca/~dborwein/cv/giuga.pdf pdf]
*Giuga, G. "Su una presumibile proprietà caratteristica dei numeri primi", "Ist. Lombardo Sci. Lett. Rend. A", 83, 511–528 (1950).