Gauss' generalisation of Wilson's theorem was given in his renowned classic, the Disquisitiones Arithmeticae (1801). In [1], Karl Dilcher and I gave the first extension of the Gauss-Wilson theorem, the key new object being what we call a 'Gauss factorial'. This entirely elementary object opens up a whole new world of interest, revealing many new results in Number Theory.
By a pleasing co-incidence this new object has important consequences for another classic (and remarkable) theorem of Gauss: his (1828) binomial coefficient (mod p) congruence, and also the closely related, and equally remarkable congruence of Jacobi (1837).The former concerns primes (5, 13, 17, 29, ...) that are 1 (mod 4), while the latter concerns primes (7, 13, 19, 31, ...) that are 1 (mod 3).
In a 1983 Paris seminar Frits Beukers conjectured a mod p^2 extension of Gauss' binomial coefficient congruence, and that conjecture was settled in 1986 by S. Chowla, B. Dwork and R. Evans. In the late 1980's, R. Evans and K. M. Yeung independently proved a mod p^2 extension of Jacobi's congruence.
No mod p^3 extension of either Gauss' or Jacobi's congruences had ever been conjectured, but in 2008 - as a side outcome of another investigation of ours [3] - we formulated and proved mod p^3 extensions of both the Gauss and Jacobi congruences [2].
Before treating the above, I shall briefly review some classic work of Fermat and Lagrange, and - should time allow - inform of some recent developments
References.
[1] John B. Cosgrave and Karl Dilcher, Extensions of the Gauss-Wilson theorem, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8, #A39, 2008.
[2] John B. Cosgrave and Karl Dilcher, Mod p^3 analogues of theorems of Gauss and Jacobi on binomial coefficients, Acta Arithmetica, Vol. 142, No. 2, 103-118, 2010.
[3] John B. Cosgrave and Karl Dilcher, The multiplicative orders of certain Gauss factorials, International Journal of Number Theory, 7 (2011), 145-171",
