From: Robin Chapman Subject: Re: A problem in elementary number theory Date: Fri, 14 Nov 1997 02:41:12 -0600 Newsgroups: sci.math In article <346ABDB3.2B57D46A@maths.ex.ac.uk>, I wrote: > > A colleague recently asked my the following simple question for > which alas I have no approach. Is there anyone out there > with any useful leads? > > Let p be a prime congruent to 3 modulo 4. Using Wilson's theorem > it's easy to show that ((p-1)/2)! = +-1 mod p. Can one say > anything about the p such that ((p-1)/2)! = 1 mod p? Kurt Foster gave me a useful hint to consider the analytic class number formula. Let R and N be the numbers of quadratic residues and non-residues respectively between 1 and (p-1)/2 respectively. Let h be the class number of the quadratic imaginary field Q(sqrt(-p)). The analytic class number formula gives h = R - N for p = 7 mod 8 and 3h = R - N for p > 3 and p = 3 mod 8. The numbers (a/p) for 1 <= a <= (p-1)/2 form a complete set of quadratic residues mod p and so their product is 1. Hence (p-1)/2! = (-1)^N mod p. Expressing N in terms of h we get that (p-1)/2! = 1 mod p iff p = 3 or P= 3 mod 8 and h = 3 mod 4. Robin Chapman "256 256 256. Department of Mathematics O hel, ol rite; 256; whot's University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no. rjc@maths.exeter.ac.uk 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn ============================================================================== Newsgroups: sci.math From: Robin Chapman Subject: Re: A problem in elementary number theory Date: Mon, 17 Nov 1997 08:29:14 GMT I wrote: > > = (-1)^N mod p. Expressing N in terms of h we get that (p-1)/2! = 1 mod p > iff p = 3 or P= 3 mod 8 and h = 3 mod 4. > Ouch squared. Make that _iff p = 3 or p = 3 mod 4 and h = 3 mod 4. -- Robin Chapman "256 256 256. Department of Mathematics O hel, ol rite; 256; whot's University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no. rjc@maths.exeter.ac.uk 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn