Numbers with anticlosed sets of non-squares

[franktaw at netscape.net]

Here's a simpler proof for this part of the result.  Start with the Pythagorean triple 9+16=25.  These are all non-zero modulo any prime greater than 5, so multiply by any non-square residue to get a non-square as the sum of two non-squares.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: franktaw at netscape.net
To: jens at voss-ahrensburg.de; seqfan at ext.jussieu.fr
Sent: Tue, 17 Jan 2006 11:43:16 -0500
Subject: Re: Numbers with anticlosed sets of non-squares


There is a well known result that, modulo any odd prime, every residue is the sum of two squares.  (The proof is fairly simple, using the pigeonhole principle.)  For p > 5, we can show that every residue is the sum of two non-zero squares.  If a + b = 1 with a and b non-zero squares, and r is any non-square, then ra + rb = r is a non-square as the sum of two non-squares.
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