Re: arithmetic progressions containing squares?

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 11/08/04 

Date: 8 Nov 2004 15:56:42 -0500

In article <cmo198$4lm$1$8300dec7@news.demon.co.uk>,
Glen Able <smDELETEecklers@hotmTHISail.com> wrote:
>Don't know if this is a common/well-known question or not and, as for so
>much maths stuff, Google is too blunt a tool to dig out the facts.

>Anyway, I noticed that 3x + 2 doesn't produce a square for any integral x.
>It's pretty simple to prove, by showing that squares can only be congruent
>(mod 3) to 0 or 1, whereas 3x + 2 is always congruent to 2.

>What about the general case of ax + b - is there a simple relationship
>between a and b that tells you if this particular arithmetic progression
>contains squares or not, or is there nothing simpler than enumerating all
>the possible values mod a? I did a computer program to investigate this but
>no obvious pattern emerged. (Incidentally, I was also a bit surprised about
>how hard it was to write an efficient 'is this integer a square' function -
>the brute force check was painfully slow, and my feeble optimisation was to
>check if the integer had suitable values mod 3, 5, 7, 11 etc.)

Check on quadratic residues. It is only necessary for something
to be a quadratic residue mod 8 or mod odd primes, assuming that
a and b have no common factors, and a little harder if they do. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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