"Victor" == Victor S Miller <victor@idaccr.org> writes:
Victor> 3) Since we're asking for the Victor> multipicative order of an element of a finite field, this is Victor> much like the question of Artin's conjecture which asks how Victor> often 2 is a primitive root (or more generally, how often a is Victor> a primitive root). The really interesting quantity is how far Victor> it deviates from the maximum value. So one would want to Victor> define fq(p) = (p-1)/fp(p) if 5 is a quadratic residue of p, Victor> and fq(p) = (p^2-1)/fp(p) if 5 is a quadratic non-residue of Victor> p. fq wouldn't be bounded, but would tend to be small. Slight correction: if 5 is a non-residue, since Norm((1+sqrt(5))/2) = -1, (1+sqrt(5))/2 is contained in a cyclic subgroup of order 2*(p+1) (the elements of norm +/- 1), so that in this case we should define fq(p) = (2*(p+1))/fp(p). As I mentioned before, 5 is special. So that Chris's data would give (for the values of fq, instead of fp): mod 3 period 8 1 mod 7 period 16 1 mod 11 period 10 1 mod 13 period 28 1 mod 17 period 36 1 mod 19 period 18 1 mod 23 period 48 1 mod 29 period 14 2 mod 31 period 30 1 mod 37 period 76 1 mod 41 period 40 1 mod 43 period 88 1 mod 47 period 32 3 mod 53 period 108 1 mod 59 period 58 1 mod 61 period 60 1 mod 67 period 136 1 mod 71 period 70 1 mod 73 period 148 1 mod 79 period 78 1 mod 83 period 168 1 mod 89 period 44 2 Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin