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**FERMAT’S LAST THEOREM: THE COMPRHEHENSIVE PROOF
RESULTING IN CALCULATING ALGORITHM.**

* *

*by Ivan M.
Obushenko, Ph.D.*

*(Ukraine)*

E-mail: obush@estart.com

*Printed on March 8-10, 2005*

The new approach for consideration of the legendary Fermat’s Last Theorem has been developed. One advantage has been received from proposed consideration: the calculating algorithm for PC – “ Obushenko-Fermat MachineÔ ” making the calculation process by fully automated and expendable unlimitedly however it had been shown that it is unnecessary, which was unexpexted. The “byproduct” of calculation is a finding of all integer triads fulfilling to the Pythagoras Theorem.

On February 25, 2005 I heard from the Ukrainian radio broadcast a presentation by Dr.Phys.-Math.Sci. on subject of the famous Fermat’s Last Theorem. She (well, what a difference, if it would be He?J) mentioned about enormous efforts made by Robert Wiles whose results were not recognized and accepted as the final proof (the same conclusion I met at reference [2]) that surprised me a lot because the acclaim made in the media about 10 or more years ago produced the impression of success on naïve public (well, on myself too). I remembered the theorem from my studies at the Senior High School (well, I had the really good teacher despite she lost to me a few “good” problems; so what? As a teacher she was very resourceful: I realized that when we’ve got a substitute man for the whole trimester…) when like anybody else I tried my “wings” on that prestigious subject too, which doesn’t seemed to be complicated on the first look and now – many years later – started again immediately on a fresh sight. Dr. Sci. summarized her lovely presentation by words about lying beyond the real world when you have deal with such thing as Fermat’s Theorem. Very scientifically and encouraging doesn’t it? It seemed to me I got some idea from air how to do it although it was not quite clear. Anyway I wrote some notes, which gave me impression that I am on right way and next day I told to my son that I did it. One more day later I’ve met the dead ends as before. But I already gave my word to my son and so I had nothing to do as go forward.

For integers *a, b,
c, *and *n*>2 a^{n} + b^{n }¹ c^{n}
.

The current consideration is substantially based on following Lemma expressing my basic idea led me forward.

__Lemma__*.* For any three
numbers *a, b, c*, for which * a *+* b **> c* some triangle can be constructed.

That *Lemma* arisen from the everyday experience and
can’t be proved in principle on my opinion. So it rather *Axiom* than *Lemma*
. Indeed, let’s make a drawing for *a, b **< c* :

*a m b
*

*Drawing 1.*

*c*

* *

Case#1. __What,
if a + b ____<____ c__

__ __

Let’s put for clarity *|a|<|b|<|c|, where |a|*=mod*
a* .

** ***b’ Drawing. 2*

*a b*

* *

*c*

*for n=3: 7^3-6^3-5^3 = 2, 9^3-8^3 -6^3=1, 12^3-10^3-9^3= -1 *

** for n=4: 6^4-5^4-5^4=46, **(by
the way from

*for n=5: 17^2 - 16^5 -13^5= -12 and *

*for c>20 and/or for n>5: the differences become larger than 1000 and
further rapidly grow in full correspondence to the just said.*