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FLT 8-19-11 6-25a



There may be a key to Pierre Fermat's conceptualization. Fermat did not assert that there is no A or B such that there can be a solution. He asserted there is no cube added to another cube such that there can be a solution. ( p265, Devlin, K., Mathematics the New Golden Age, 1999, p265 .) Fermat's point of departure was A^ N^ and B^ N^. To work from Fermat's perspective one has to let go of the question, is A or B irrational.

The construction here begins by defining A as always rational. The expansion of any rational A always produces a rational A ^N^. Pierre Fermat worked with rational numbers, I.e. , with A^N^+ B^N^ = 1, rather than A^N^+B^N^ = C^N^ (Devlin, ibid, p282.) Since B ^N^ =(1- A^N^), then, by substitution A^N^+ B ^N^= 'A^N^+ (1-A ^N^' )= 1. By constructing this problem with A always rational, which always produces a rational A^N^ , then (1- A^N^) is also always rational. For any rational A and for any N>2, these three knowns are always present, are always rational, and would always follow the same principles.

Since the A's are the only possible rational numbers, the expansion to any A^N^'s yields the only possible rational quantity. For (1-A^N^) to be a part of a solution it has to have the same quantity as one of the A^N^ 's. The Fermat question can be expressed most simply as, for any N>2, can a (1-A^N^) have the same quantity as a A^N^? This construction completely avoids the nature of B. There is no need to prove that B is always irrational.

One proof Fermat had written was 2 ^N^-2 gives a number that can be divided by that N , also written 2 ^N^-2 modN (Stark, H.M., An Introduction To Number Theory, 1994, MIT Press.) 2^ N^-2 is also A ^N^-2 when A = 2 of A ^N^+ (1-A ^N^) = 1, The way the equation is constructed makes a difference in exploring the issues of the problem. For instance, using the A ^N^+ (1-A ^N^) = 1, organizes and simplifies the information such that it is easier to observe essentials, possibilities and alternatives. Constructing an approach that begins with a rational A, a rational A ^N^, and a rational (1- A ^N^), then for any N>2 these three knowns are always present, are always rational, and would always follow the same principles. For all rational sequences of A (0,1), i.e.,{.1,.9}, ... {.000...1,.999...9}, every sequence begins with a “1” as .1, .01, .001, .0001, ... .000...1, and this “1” to any power of N creates an A ^N^ with a “1” as the only digit. No matter what the A^N^ is, the “1” is the begining of the sequence. The next in the sequence is a “2”, as .2, .02, .002, .0002, ... .000...2, and this “2” to any power of N creates an A ^N^ with Fermat's 2 N.



P.K Tam, in the Southeast Asian Bul of Mathematics v32 , 2008, p1177-81, presents proofs that Fermat's Last Theorem can be understood in “a self-contained elementary and purely algebraic treatment” (p1177). This article has not been discussed in the West. His paper supports the present thesis, which is a “self-contained elementary and purely algebraic treatment.” Another paper ignored is E. E. Escultura's “An Exact Solution of Fermat's Equation ” Non-Linear Studies v5, 1998, p227-255, in which he demonstrates that “the loss of certainty affects all concepts and propositions involving... infinite spaces.” The present approach constructs this problem with A always rational, which always produces a rational A^N^ , then (1- A^N^) is also always rational. For any rational A and for any N>2, these three knowns are always present, are always rational, and would always follow the same principles--and never involves “infinite spaces.” This paper is on the side of Tam, presenting a simple construction that avoids the pitfalls of infinite spaces and has no constraints due to “loss of certainty.” Escultura's telling criticism contributes to a construction avoiding “pitfalls of infinite spaces.

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