Variations on Wilson's theorem
Vortrag von Prof. Dr. Michele Elia
Datum: 13.05.14 Zeit: 14.30 - 15.30 Raum:
Abstract:The factorial w(p)= ((p-1)/2))! mod p provides a square root of (-1)^((p+1)/2) as a consequence of Wilson's theorem. Thus, two sequences {w_1(p)} and {w_3(p)} of ±1s can be defined for primes congruent 1 and 3 modulo 4, respectively, as follows
1) if p is congruent 1 modulo 4, let w_1(p) = w(p) x/ y be an element of the first sequence, with x and y satisfying x^2+y^2=p. w_1(p) is either 1 or -1 since both w(p) and x/y are square roots of -1 modulo p.
2) if p is congruent 3 modulo 4, let w_3(p) = w(p) be an element of the second sequence. w_3(p) is either 1 or -1 since w(p)^2=1 modulo p. In both sequences {w_1(p)} and {w_3(p)}, the numbers 1 and -1 apparently alternate infinitely many times in unpredictable ways.
With these premises, several observations on properties of the two sequences {w_1(p)} and {w_3(p)} and their relations to binary quadratic forms with negative discriminant are reported. In particular, this link points out a notable connection of the relative densities of 1s and -1s in each sequence with the Gauss class number problem for imaginary quadratic fields.