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Topic: short Proof of how Fermat made his FLT discovery plus some additional material.
Replies: 11   Last Post: Aug 19, 2011 10:36 AM

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Posts: 116
Registered: 4/28/10
Re: short Proof of how Fermat made his FLT discovery plus some additional material.
Posted: Aug 19, 2011 10:36 AM
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> That's a pity, but very soon something else
> will be rediscovered:
> FLT really short proof like used to be (by
> P.Fermat):
> The X;Y;Z integers could be substituted with
> few parameters. For n=3 proportionally raised
> parameters could establish proportional X'=k*X;
> Y'=k*Y; Z'=k*Z; but from n=5 such operation
> is not possible: proportional gain of parameters
> can not be expressed with algebraic numbers for
> to achieve X';Y';Z'; as integers therefore
> parameters
> in part or whole are irrational numbers !
> Q.E.D.
> How we can establish such proper parameters
> and for more details see my topic:
> "Second or not visible side of FLT ?"
> Best Regards
> Ro-Bin

A key to Fermat's Last Theorem is presented. Fermat worked with rational numbers, which has the equation A ^N+ B N = 1,which by substitution is A^ N+ (1-A^ N)=1. The construction begins with the A as always rational. It follows that A^N and (1-A^N ) are also rational. Rational sequences of A are {.1,.9}, {.01,.99}... {.00...1,... .99...9}. The rational A's are by definition the complete set of rational digits. A^N is the only possible rational expression of A. To be a solution, (1-A^N) has to have the same quantity as one of the A^N . Most simply, can a (1-A^N) have the same quantity as one of the A^N ?
Fermat gave a proof that 2^N-2 gives a mod N for N that are prime. There is an exact parallel in derivatives of A^N and of (1-A^N), the 2^N-2 modN appearing in the left corner of the column of d(2). The right corner also has d(N) that are modN. A rationale for the construction of all derivatives for all A and all N is presented. Also, the simple construction has A^N 's that only can have a finite number of possiblities, offering a simple support to Falting's proof of the Mordell conjecture of 1922 . A contribution here is to reopen the discussion, did Fermat have a conceptualization re A^N and B^N that is worthy of investigation? He may have had a simpler way of conceiving that can be utilized in contemporary questions.
Martin E Johnson, PhD

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