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