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^

**= 1,**

^{N^}**rather than A^**

^{ }**B^**

^{N^}+*= C^*

^{N^}*(Devlin, ibid, p282.) Since*

^{N^ }*B ^*, then, by substitution

^{N^}^{ }=(1- A^^{N^})*A^*. By constructing this problem with A always rational, which always produces a rational A^

^{N^}+ B ^^{N^}= '**A^**^{N^}+ (1-A ^^{N^}'^{ })= 1**, then (1-**

^{N^ }^{ }A^

*is also always rational. For any rational A and for any N>2,*

^{N^})^{ }*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^

**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**

^{N^ }'*, 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*Stark, H.M.,

^{ ^N^}-2 modN (*An Introduction To Number Theory*, 1994, MIT Press

*.)*2^

^{ N^}-2 is also A

^{ ^N^}-2 when A = 2 of A

^{ ^}

^{N^}

**+**(1-A

*The way the equation is constructed makes a difference in exploring the issues of the problem. For instance, using the A*

**) = 1,**^{ ^N^}^{ ^N^}

**+**(1-A

*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*

**) = 1,**^{ ^N^}^{ ^N^}, and

**a rational**

**(**1-

^{ }A

*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*

^{ ^N^}),*(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***with a “1” as the only digit. No matter what the A^*

**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**

^{N^ }*with Fermat's*

**A**^{ ^N^ }*.*

**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^

*is also always rational. For any rational A and for any N>2,*

^{N^})^{ }*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.*”