IN DEFENSE OF MR. FERMAT
This brief paper constitutes a second version of a paper formerly listed under the same name. The concern, of course,is the contradiction in terms of Fermat's Last Theorem. The first document contained irrevelant material the purpose of which to expand the subject text at the expense of motivation and purpose. What remains here is a matter of logical immendiacy.
Definition and Constraints
(1) r^n = a^n + b^n ,
where a,b and r are positve with n>2, where ^ denotes exponentiation.
Rewriting Exponents
What should be considered is the prominent issue is the exclusion of exponent 2 at definition of variables.This value must have exceptional properties in the context of eventual generalization and consequent proof. The method is described as "Rewriting Terms."
In order to define and exposnent n as a positve exponential base as positive, nothing intrinsic can contradict the "definition ," as such. the exponent 2 does precisely that.
OBJECT: [(1) holds for any n such that n is greater than or equal to 3] implies (1) 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 > 0 ] is necessary.
CASE 1: n has an odd divisor, q, greater than 1.
Let (1) 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, the fact that plus or minus r, plus or minus a and plus or minus b are primitive of necessarily positive bases, [v, u, w] since the latter are 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,n’] has strictly positive exponential bases.
CASE 2: n = 2t where t is greater than or equal to 2. Let (1) be written;
r[2*(t-1)*2]= a[2* (t-1)*2] + b[2* (t-1)*2] (where * represents multiplication.)
which upon substitution becomes v2 = u2 + w2. As before, the fact that 2*(t-1) is even and the existence of roots plus or minus r, plus or minus a and plus or minus b, corresponding to [v,u,w] which are substituted for even exponentials imply these variables are all positive. Note that the case n=2 cannot be rewritten such that its root is not possibly negative.
Having limited the existence of a positive-only solution for a rewrite of (1) such that [r,a,b,n] goes to [v,u,w,n’] for all n greater than or equal to 3, order is specific and consequently subject to contradiction (absent of n=2.)
Limitation of Trichotomy
Suppose now that (1) has been rewritten for some n, where n is equal to or greater than n’ so that [r,a,b and n] are transformed to a reduced form, redefining [r,a, b and n] to be [u,v,w and n’]. Thus n’ is greater than or equal to 2 implies [v>u>w>0]. Consider the transformation (under n’), T, such that T:v to v, u to w, and w to u. Respecting (1). T(v,u,w) = Tnot(v,u,w) where Tnot signifies “not T”. It follows that [u>w] can be transformed to [w>u] such that [u>w] is concurrent.! This can happen only for the case, u = w, which is moot. Conclusively, additive order must be assignable when it exists. ( ! denotes contradiction.).
QED
Kerry M. Evans
ON THE FULL BEAL CONJECTURE
The purpose here is to confirm Beal’s Conjecture, specifically as a consequence of Fermat’s Last Theorem (FLT) and a defining transform, T:, used to generalize the problem, all in context with conditions necessary to uniquely describe the values of all supposed solutions.
Consider the sum of a and b such that a>0, b>0 and p>0 where a and b are solution pairs (prior to further specification).
a + b = pq.
a and b can be “cross-defined,” (one in terms of the other) by the transform T: a to bm, b to an, with m,n and q, all greater than 2, uniquely defining all Integer contributors.
Applying T,
T1(a + b) = bm + an = pq,
which is the general representation of Beal’s Conjecture when equivalence on the right is negated.
T, once incepted, may be repeated as Tz such that z is an element of {2, 3, . . . }, where continued application neither extends or restricts the validity of its first application, subject to the same limitations.
Reapplying T:,
T2(a + b): = T(bm + an) :
bmn + amn = pq ,
which is a broad rendering of contradiction to FLT, i.e. q = mn is possible.
For simplification relative to FLT, x and y are substituted to yield:
xm + ym= pq,
best described as “the sums of like powers that add to an exponential.” The negation of FLT lies within the range of this generalization. Unless a = b , the expression is false (or rather an inequality) consequent to recent proof of FLT (see: WWW). Contradiction of the equivalence negates T2(a + b), a direct implication of T1(a + b). Reasoning by nature of contrapositive; “ bmn + cmn unequal to pq, a transformation under T:, implies bm + an unequal to pq , under the same transform,” which initially preceeded it. Thusly, the inequality of T is true over all its application. Conclusion: the Beal Conjecture is true.
The object has thus been satisfied except for when pq is a power of 2, x=y=2^m, in which case q must be reduced to an equivalent 22 + 22 = 23, which is an excepted case; and further, unless a common divisor is present, in which case factorization alters the general representation of the problem.
QED
Suggested reading: “In Defense of Mr. Fermat,” on WWW at fermat.yolasite.com.
By same author: “An ‘Expository’ Proof of the Goldbach Conjecture,” at goldbach.yolasite.com.
email: kerrymerry2000@yahoo.com .
