FLT false in Q(Sqrt(5))

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1 total messages Started by Paul Bruckman Sun, 28 Jun 1998 00:00

#1255

Author: Paul Bruckman
Date: Sun, 28 Jun 1998 00:00

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I posted a similar question a while back, but never received a reply, so
I'll try again.  Fermat's Last Theorem is false if we extend the domain
from integers to integers in Q(Sqrt(5)).  For example, if we let
a = (1+Sqrt(5))/2 , b = (1-Sqrt(5))/2, we may easily verify that
(a+4)^3 + (b+4)^3 = 6^3 .  How many more counterexamples like this one
can we find (using exponents of 3 or more)?  It may be proven that
(xa+y)^4 + (xb+y)^4 = z^4 has no solutions in integers x,y and z, and
neither does the comparable equation if we replace z^4 by 25z^4 . There
are trivial solutions, namely x = 2u, y = -u , z=0 in case the exponents
are ODD . What sort of generalizations are possible from this?  I am
primarily interested in Q(Sqrt(5)).  Would be interested in starting a
dialogue on this.  Paul Bruckman

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