In Defense of Mr.
Fermat
The
concern here, of course, is the contradiction in terms of Fermat’s Last
Theorem. Many attempts and supposed proofs have preceded
by professional Mathematicians and amateurs as well. Invariably, these proposed proofs have met
with opposition by nature of inherent dubious assumptions and illicit
conclusions. Thus clarity is a matter of
first order. Motivation is equally
intended. Hopefully, what remains is a
matter of logical immediacy.
Definition and
Constraints *
rn =
an + bn ,
where
a, b and r are positive with n>2.
Rewriting Exponents
What should be considered as the prominent issue is the exclusion of exponent 2
at definition of variables. This value has exceptional properties in the
context of eventual generalization and subsequent proof. The method is
described as “Rewriting Exponents.”
OBJECT: [ * holds for any n
such that n is greater than or equal to 3] implies * may be rewritten
(in possibly different r, a, b and n) as
vn’ =
un’ + wn’,
where
n’ is greater than or equal to 2 such that v, u and w are necessarily
positive.
CASE 1: n
has an odd divisor, q, greater than 1.
Let * be
written, r[(n/q)·q] = a[(n/q)·q]
+ b[(n/q)·q] and substituted as vq
= uq + wq.
If n/q is even, v, u, and w, are positive
substitutions of an even exponent. On the other hand, if n/q is odd, specified
positive conditions, a>0 and b>0, imply the relation, in v, u, w and n’
has strictly positive exponential bases.
CASE 2: n =
2t where t is greater than or equal to 2. Let * be written; [exp denotes “to the exponent of”]
r exp(2∙2(t−1))= a exp(2∙2(t−1))
+ b exp(2∙2(t−1))
which
upon substitution becomes v2 = u2 + w2. As
before, the fact that 2(t-1) is even implies v, u, and w are
positive substitutions for even exponentials.
Note that the case n=2 cannot be non-trivially rewritten such that its
root is not possibly negative.
Having
limited the existence of a positive-only solution for a rewrite of * such that r, a, b, and n goes to v, u, w, and
n’ for all n greater than or equal to 3, order is specific and consequently
subject to contradiction (absent of n=2).
Violation of Order
Suppose now
that * has been rewritten so that r, a, b and n are transformed to a reduced
form to be u, v, w and n’. Thus rewritability and initial conditions imply, n’
≥ 2 and v, u and w are positive to exclude the case n = 2. Consider the transformation (under n’), T,
such that T:v to v, u to w, and w to u. T(v, u, w) = Tnot(v, u, w) where Tnot
signifies “no application of T”. It follows that [u>w] can be transformed to
[w>u] such that [u>w] is concurrent!
( ! denotes contradiction.) This can happen
only for the case, u = w = 2, which is moot. Conclusively, additive order must
be assignable when it exists. QED
Beal: http://beal.yolasite.com
Goldbach: http://goldbach.yolasite.com
