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 which is now unspecified: a + b = ?
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 and n greater than 2 subject to the specification pq where q > 2.
T1(a + b) = bm + an
= pq
which
is the general representation of Beal’s Conjecture when equivalence on the
right is negated.
T
may reapplied such that it does nothing to change the conditions on a and b. Reapplying
T:
T2(a + b): = T(bm + an) :
bmn + amn = pq,
The latter is possible only when a = b = 2 (see:
http://fermat.yolasite.com),
in which case, 2mn + 2mn = 2mn+1. Now let 2m + 2n = pq and
further, m > n where a = b = 2. Then
2m-n + 1m-n = pq / 2n.
Thus, the
exponential bases on the left are unequal and therefore the quantity on the
right can be at most a square when (m–n) ≥ 3. (again
see: http://fermat.yolasite.com).
Otherwise, 2r + 2r = 2r+1 which is factorable to 1z +
1z = 2 where a = b = 1 while pq
is reduced to 2.
In all other cases, a, b
and p are factorable such that the general representation fails to hold.
Incidentally,
Catalan’s Conjecture is also easily rendered using the same precepts presented
here.
Fermat: http://fermat.yolasite.com
Goldbach: http://goldbach.yolasite.com
.
