Re: For Fermat fans

From: William Elliot (marsh_at_privacy.net)
Date: 03/09/05 

Date: Wed, 9 Mar 2005 04:08:29 -0800

From: Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk>
Newsgroups: sci.math
Subject: Re: For Fermat fans

> William Elliot wrote:
> > On Tue, 8 Mar 2005, Larry Hammick wrote:
>
> >> Show that any positive integer which is congruent to 36 mod 40
> >> is the sum of four squares all of which are 9 mod 40.
> >> The first solver wins a copy of the complete works of JSH.
>
> > 36 = x^2 + y^2 + z^2 + t^2 (mod 40)
> > x^2 = y^2 = z^2 = t^2 = 9 (mod 40)
> > x, y, z, t = 3,37 (mod 40)

> 7^2 = 49 = 9 (mod 40).
> Try again!

(x - 3)(x + 3) = 0 (mod 40)
x - 3 = 1, 2, 4, 5, 8, 10, 20, 40
x + 3 = 40, 20, 10, 8 5, 4, 2, 1

     x = 4, 5, 7, 8, 11, 13, 23, 3
     x = 37, 17, 7, 5, 2, 1, 39, 38

   x = 3, 7, 13, 17, 23, 27, 33, 37
x^2 = 9, 49, 169, 289,
x^2 = 9 (mod 4)

36 = 9 + 9 + 9 + 9
> NB 76 = 49 + 9 + 9 + 9.
116 = 49 + 49 + 9 + 9
156 = 49 + 49 + 49 + 9
196 = 49 + 49 + 49 + 49
236 = 169 + 49 + 9 + 9
276 = 169 + 49 + 49 + 9
316 = 169 + 49 + 49 + 49
     = 289 + 9 + 9 + 9
356 = 289 + 49 + 9 + 9

Whew, I don't win. ;-) 

----