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Article #1255FLT false in Q(Sqrt(5))
From: Paul Bruckman
Date: Sun, 28 Jun 1998 00:00
Date: Sun, 28 Jun 1998 00:00
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1742 bytes
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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X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: 10d67e,715efd2fccc38c3b,start X-Google-Attributes: gid10d67e,public X-Google-Thread: 107079,715efd2fccc38c3b,start X-Google-Attributes: gid107079,public X-Google-Thread: 109d8a,715efd2fccc38c3b,start X-Google-Attributes: gid109d8a,public X-Google-Thread: 10d29f,715efd2fccc38c3b,start X-Google-Attributes: gid10d29f,public From: Paul Bruckman <aardwolf@gte.net> Subject: FLT false in Q(Sqrt(5)) Date: 1998/06/28 Message-ID: <6n6ees$eql$2@news-2.news.gte.net> X-Deja-AN: 367002652 Content-Transfer-Encoding: 7bit Approved: Daniel Grayson <dan@math.uiuc.edu>, moderator for sci.math.research Mime-Version: 1.0 Reply-To: aardwolf@gte.net Originator: dan@orion.math.uiuc.edu Content-Type: text/plain; charset=us-ascii Organization: gte.net X-Auth: 570FC481129C4883514C0412 Newsgroups: sci.math.num-analysis,sci.math.research,sci.math,aus.mathematics