From: hb3@aixterm6.urz.uni-heidelberg.de (Franz Lemmermeyer)
Newsgroups: sci.math
Subject: Re: Quartic residues for p=8*k+1
Date: 9 Nov 1995 09:45:19 GMT

In article <47pmhb$gai@nyx10.cs.du.edu>,
          colin@nyx10.cs.du.edu (Colin Plumb) writes:

|> But for that, I need to know if 2, which I already (assuming that n is
|> prime) know is a quadratic residue, ie. 2 == t^2 (mod n) for some t,
|> is also a quartic residue, i.e. 2 = t^2 = s^4 (mod n) for some s.
|> 
|> I've stared at the bit patterns of a few small primes of the form
|> p = 8*k+1, and tried to discern a pattern, but nothing is terribly
|> obvious yet.

1. If you want to see a pattern, look at the representations of
   p as a sum of two squares, i.e. p = a^2 + 16b^2.

2. The resulting pattern (an old conjecture of Euler, first 
   proved by Gauss) can be proved quite elementary (Dirichlet
   was the first to do so); it can be found in quite a few texts
   on Number Theory.

3. If you want to learn about quartic reciprocity, look at
   Ireland-Rosens introduction to number theory.

franz

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