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English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.

The pre-eminent French lawyer and amateur^{1} mathematician, Advocate―Pierre de Fermat (1607-1665) in 1637, famous in the margin of a copy of the famous book Arithmetica which was written by Diophantus of Alexandria (~201 - 215 AD), wrote:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.”

In the parlance of mathematical symbolism, this can be written succinctly as:

where the triple

Rather “notoriously”, it stood as an unsolved riddle in mathematics for well over three and half centuries. Many amateur and great mathematicians tried but failed to prove the conjecture in the intervening years 1637- 1995; including three of the World’s greatest mathematicians such as Italy’s Leonhard Euler (1707-1783), France’s Pierre-Simon, marquis de Laplace (1749-1827), and the celebrated mathematical prodigy, genius and Crown Prince of Mathematics, Germany’s Johann Carl Friedrich Gauss (1777-1855), amongst many other notable and historic figures of mathematics.

Without any doubt, the conjecture or Fermat’s Last Theorem is in-itself―as it stands as a bare statement, deceptively simple mathematical statement whose any agile 10-year-old mathematical prodigy can fathom with relative ease. As already said, Fermat famous―via his bare marginal note stated he had solved the riddle around 1637. His claim was discovered some 30 years later after his death in 1665, as an overly simple statement in the margin of the famous copy Arithmetica. As it is well known from the history of mathematics, Fermat wrote many notes in the margins and most of these notes were “theorems” that he claimed to have solved himself. Some of the proofs of his assertions were found. For those that were not found, all the proofs save for one resisted all intellectually spirited efforts to prove it and this was the marginal note pertaining the so-called Fermat’s Last Theorem.

This marginal note dubbed Fermat’s Last Theorem, was the last of the assertions made by Fermat whose proof was needed, and for this reason that it was the last of Fermat’s statement that stood unproven, it naturally found itself under the title “Fermat’s Last Theorem”. Because all of the many of Fermat’s assertions were eventually proved, most people believed that this last assertion must―too, be correct as Fermat had claimed. Few― if any, doubted the assertion may be false, hence the confidence to call it a theorem. Simple, the proof Fermat claimed to have had, had to be found! Alas, reality could prove otherwise that the proof was not a mere summer walk in the park.

So the question is: Did Fermat actually posses the so-called “truly marvelous” proof which he claimed to have had? This is the question many have justly and rightly asked over the years and this reading makes the temerarious endeavour to vindicate Fermat that he very well might have had the “truly marvellous” proof he claimed to have had and we accomplish by providing a proof that employs elementary arithmetic methods that were available in Fermat’s day.

Surely, there are just reasons to doubt Fermat actually had the proof and this is so given the great many notable mathematicians that tried and monumentally failed and as-well, gave the number of years it took to find the first correct proof. The first correct proof was supplied only 358 years later by the English Professor of mathematics at the University of Cambridge―Sir Professor Andrew John Wiles, in 1995 [

To add salt to injury i.e. add onto the doubts on whether or not Fermat actually had his so-called “truly marvellous” proof is that Sir Professor Andrew Wiles’s proof^{2} employs highly advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Actually, these tools and methods were invented (discovered) in the relentless effort to solve this very problem. Herein, we supply a very simple proof of Fermat’s Last Theorem.

That said, we must hasten to say that, as a difficult mathematical problem that so far yielded only to the difficult, esoteric and advanced mathematical tools and methods of Sir Professor Andrew Wiles―Fermat’s Last Theorem, as any other difficult mathematical problem in the History of Mathematics, it has had a record number of incorrect proofs of which the present may very well be an addition to this long list of incorrect proofs. In the words of historian of mathematics―Howard Eves [

“Fermat’s Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs has been published.”

With that in mind, it allows us to say, we are confident that the proof we supply herein is water-tight and most certainly correct and that, it will stand the test of time and experience.

As stated in the ante-penultimate above, in this rather short reading, we make the temerarious endeavour to answer this question―of whether or not Fermat actually possessed the proof he claimed to have had. We accomplish by supplying a simple and elementary proof that does not require any advanced mathematics but mathematics that was available in the days of Fermat. Sir Professor Andrew Wiles’s acclaimed proof, is at best very difficult and to the chagrin of they that seek a simpler understanding―the proof is nothing but highly esoteric. The question thus “forever” hangs in there to the searching and inquisitive mind: “Did Fermat really possess the proof he claimed to have had?” The proof that we supply herein leads us to strongly believe that Fermat might have had the proof and this proof most certainly employed elementary methods of arithmetics!

Before we go into the main business of the day, we shall give a short history of some notable efforts in finding a proof to Fermat’s Last Theorem. As is well known, the case for

Fermat was the first to provide a proof for the case

The case

If

The above statement is clearly evident and needs no proof. However, below we demonstrate that this statement is true. This demonstration does not constitute a proof.

What this statement really means is that the number

written as a sum or difference of two numbers p and q where

since one can always find some

in which case we will have

Setting

Proof.

If

where

Now,

where

Now, the proof that we are going to provide of FTL is a proof by contradiction and this proof makes use of Lemma §(3) whereby we demonstrate that the triple

to be true for some piecewise co-prime triple

1) We must realise that if just one of the members of the triple

then, all the members of this triple will be greater than unity i.e.

2) By way of contradiction, we assert that there exists a set of positive integers

3) If the statement (8) holds true, then―clearly; there must exist some

4) According to the Lemma §(3), the equation

5) From (10), it is clear that

Alternatively, according to the Lemma §(3), the equation

Again, from (11), it is clear that

Therefore, by way of contradiction, Fermat’s Last Theorem is true since we arrive at a contradictory result that

If the proof we have provided herein stands the test of time and experience, then, it is without a doubt that Fermat’s claim to have had a ‘truly marvellous’ proof may very well resonate with truth. The proof provided herein is not only simple, but surprisingly simple, so simple that one wonders how great mathematicians would have missed this. All this simplicity is embodied in Lemma (3). As we anxiously await the World to judge our proof, effort and work, we must―if this be permitted at this point of closing, say that, we are confident that―simple as it is or may appear, this proof is flawless, it will stand the test of time and experience. It strongly appears that the great physicist and philosopher―Albeit Einstein (1879-1955), was probably right in saying that “Subtle is the Lord. Malicious He is not”. Because in Lemma §(3), there exists deeply embedded therein, a subtlety that resolves and does away with the malice and notoriety associated with Fermat’s Last Theorem in a simpler and truly marvellous and general manner.

Given that the method used here to prove Fermat’s Last Theorem is so elementary, it is very much possible that Fermat actually processed the correct proof.

Golden GadzirayiNyambuya, (2016) On a Simpler, Much More General and Truly Marvellous Proof of Fermat’s Last Theorem (I). Advances in Pure Mathematics,06,1-6. doi: 10.4236/apm.2016.61001