We are approaching the fifth anniversary on-line of the author’s extraordinary “bare hands” proof that the sum of two natural numbers raised to the power n can never itself be an nth power of a natural number (when n>2).

When it was sent to the Mathematical Gazette in 2019 no mistake could be found in this elementary reasoning by a first pair of referees. Nor by a second set of referees, who were subsequently asked to look at it.  Later the Editor of the Mathematical Gazette himself, an accomplished mathematician, couldn’t find one either. But he turned it down anyway.

This was hardly a professional decision, because maths is a rational discipline and vague feelings don’t count as grounds for dismissing mathematical results.

We are especially fortunate in that we now know that the conjecture itself is correct, thanks to the groundbreaking work of  Prof. Andrew Wiles, who proved it for the first time in 1995 —using modern advanced (hyper-abstract) algebra. Unfortunately the advanced methods used by Wiles were far beyond the comprehension of the best mathematicians of Fermat’s day, or indeed of today’s average mathematically-educated person.

So Wiles’ groundbreaking proof  belongs to a very limited in-house elite. It can’t be understood by the average mathematically educated person. This is not a very satisfactory situation, especially since maths has been virtually cancelled in the rhetoric of Silicon Valley. Let’s face it:  it has been acutely puzzling that, for more than 350 years, an elementary proof which Fermat himself claimed to have found has not been re-discovered.

Surely this missing proof  should have been treated as being worth searching-for? Why did so few researchers wonder how Fermat managed to pull the trick? Surely Fermat’s claim must be taken seriously?  He would hardly risk losing his high reputation as a leading maths guru, by making such a distinctive claim, if he was aware it was dodgy.

After Fermat’s death, a Working Party should surely have been set up to re-think the approach he must have followed. It should have been tasked to find the answer —however long it took… Why?  Because the honour of mathematics was on the line.  Instead, the mathematic community lamely accepted “that it was lost for good”.

So, shamefully, an unconfident,  unspoken attitude has dominated for four centuries. It is that the elementary methods used by Fermat <<must have been incapable of unlocking this extremely difficult dilemma>>. It implies that Fermat’s claim <<must have been false>>.  His elementary proof must have harboured a <<mistake>>. Here, again, an unprofessional tail has managed to wag the dog of mathematics. This feeble attitude dominated for more than 350 years.  It is essentially disrespectful of the power of elementary reasoning in maths. It confirms that too many professional mathematicians have decided, over four centuries, to favour slick, high-powered, generalised methods of analysis, rather than face the hard work involved in cultivating the intense conceptual analysis which can tease-out the precise implications of elementary truths.

This attitude under-sells the conceptual power of maths.  It also incidentally pushes today’s leading-edge pure maths a long way into the distance… a very long way away from anything the average mathematically-literate person could possibly understand. This means that maths’ unique role as the <<Widely Recognised Heartland of Truth>> has been thrown away. The great maths discoveries of Antiquity (that the square root of 2 cannot be a fraction, and that there is an infinity of primes) embody recognised intellectual powers of the highest order:  because there are millions of people, including scientists and philosophers, who can appreciate the uncanny magic of the logical reasoning which they deploy. This can be (=used to be) a major source of rational thinking across society.

Since the arrival of computers to automate maths, the default position has been that these paradigmatic Euclidean theorems constitute the main social justification for so-called “pure” mathematics. The value of pure maths from society’s (and a layman’s) point of view is that it inspires creative reasoning in society. But the latest hyper-abstract results won’t inspire anyone —other than in-house gurus—  if it is based on mazy, self-referential concepts and problems a million miles away from anything the average mathematically literate person could ever hope to understand.

Today we are living through an exceptionally dark night of irrationality, usually blamed on dozy World Leaders. But there were also eight, hushed-up, major blunders the maths establishment made during the 20^th century.  (See my essay in the Nov. 2022 New English Review.) Faith in reasoning and intellectual endeavour are on the floor.  Unless we make intense efforts to resuscitate them, they will stay there.

So today’s deliberate neglect of results achieved by “bare hands” methods cannot be justified. They should be the maths community’s no. 1 priority. Otherwise it is going to be widely thought that the leading gurus of higher maths are an in-house elite which has decided to “do its own thing” and, in effect, ignore the rest of humanity.

The present author decided in 1991 to embark on a bare hands analysis of the Fermat problem.  He had no knowledge of the latest hyper-abstract algebra or number theory.  This was his special advantage: it meant that he would not be tempted to stray into modes of thought which could not possibly have been in the mind of Fermat. Only concepts roughly equivalent to that of specialist maths A-level would be involved.

His only resource was an obstinate determination to proceed… until it became clear either that Fermat’s conjecture was true, or that there were counter-examples. His strategy was to spend about an hour a day on this project. He originally imagined that this might take five years…

In the end it took him more than 10,000 hours, and 29 years to reach the conclusion he posted on-line in July 2020.  This seemed to be the only way to bring the dramatic conclusion of this historic research to the attention of curious maths-literate people everywhere.  But then the Mathematical Gazette decided not to print his exposition. They based their decision on the weeze: <<We can’t find a mistake, but there must, surely, be a slip-up somewhere>>.

A PRIZE OFFER

This is not a satisfactory situation.  To mark the 5-year anniversary of this unchallenged on-line exposition, the author has decided to offer a reward of £3,000 to the first person who can find a self-evidently exposition-derailing mistake in the on-line argument. He or she will have to recruit   two independent colleagues to check and confirm his or her claim —they will each receive £1000.

 The exposition is on blogs 1 and 2 of this website.

The offer expires on July 1^st 2025 and if no disproof has been forthcoming, the author intends to adopt the default assumption that his reasoning <<must be correct>>.  Send your name, disproving reasoning, and confirming colleague names,  to:  per4group@gmail.com . CHRISTOPHER ORMELL around April 1^st 2025.