Set Theory and Logic


There Are Infinitely Many Theorems as Difficult to Prove as FERMAT’S Last Theorem: a Characterization of Such Theorems

Authors: Allen D Allen

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a change from > to < rendering their transitive natures unavailable. A related, self evident symmetry illustrates another problem that would have plagued Fermat and centuries of successors. FLT asserts such a narrow proposition, it is difficult to find an antecedent while easy to find a non equivalent consequence. For example, if FLT asserted that the exponent n is even, then FLT would be equivalent to the proposition that Fermat’s equation has two solutions, one for positive bases and one for their negative counterparts. This could be addressed with conservative transformations. The example provided by FLT motivates the use of an early paper by the author to prove a theorem on theorems. The theorem on theorems demonstrates there are infinitely many theorems as difficult to prove as FLT.

Comments: Abstract contains 200 words, ms runs 6 pages

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Submission history

[v1] 2016-04-05 09:40:07
[v2] 2016-04-10 18:42:42

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