It is now exactly three years since the conclusion of my project to explain the validity of Fermat’s Conjecture by elementary methods was put online and also published by Ingleside Ashby in a small cottage edition. Fermat had claimed in a marginal note in a book found after his death that he had proved that xn + yⁿ can never equal zⁿ when n>2 and n, x, y, z are natural numbers.  In other words that

                              xⁿ + yⁿ  =  zⁿ  is never true……………(1)

This may be called ‘Fermat’s Conjecture’.

In my opinion it is a scandal that this lost proof has been accepted as <<just one of those things>>. A task force should have been convened in the 18^thcentury to “do what it takes” to reconstruct Fermat’s amazing argument.

I started my project to try to re-discover this frustratingly lost proof in January 1991 and it took me 29 years to complete the quest. I tried to spend at least an hour a day on the project, and actually exceeded that on many occasions. Much of the work consisted of conceptual analysis and thought experiments —probing to find what obscure obstacle was preventing the LHS of (1) from equalling the RHS. So in total I spent at least 10,000 hours trying to pin Fermat’s conjecture down to a simple, understandable, elementary, unambiguous form, where it could be analysed using the kind of maths available to Pierre de Fermat in the 17^th century.

[This mainly consists of the algebra of binomial expansions which we teach to 16-year olds, plus some simple divisibility logic.]

I started the project in Canada in 1991 with no baggage whatever, because my knowledge of number theory was nil.  I figured that my state of total ignorance of abstract modern thinking would be an advantage… if I were to be fortunate enough to stumble onto a successful line of elementary reasoning. Four years after I began, Andrew Wiles pulled off the greatest mathematical coup in the history of the subject by proving Fermat’s Conjecture (1) above. What he found, though, was definitely not the proof Fermat had devised (assuming his proof was valid). We know Wiles’ proof used hyper-abstract modern concepts far beyond the state-of-the-art-knowledge available to Fermat..

Wiles’ impressive coup has been, in many ways, a helpful development… It established the result as being valid. It has also removed at a stroke much of the taboo surrounding Fermat, namely that it was considered by many mathematicians before 1995 to be a stamping ground for manic amateurs. It was a waste of time to give close studious attention to their shaky arguments, which invariably rested on crass manipulative mistakes.

So was it, in retrospect, worth spending more than 10,000 hours heavy lifting on this quest?

Yes, because the analysis which has finally emerged has several worthwhile aspects.  It’s form is, of course, to disprove the “nominal hypothesis” that the LHS of (1)  above can equal the RHS.

Features which made the difference:

1 A switch of the hypothesis from the n, x, y, z format above in (1) to the new form in (2)

                    (a+h)^p + (b+h)^p =(a+b+h)^{p  ………………………..}(2),

where a, b, h are natural numbers and p is a prime number > 2.

This alone is a great step forward.  Because the LHS is obviously smaller when h is small, and obviously larger when h is large, we are immediately aware that there

Is a palpable problem here: how does the LHS manage to pull off this summersault as h increases?

Now we can address the $64 question: Is it ever possible for the LHS of (2) to equal the RHS?

[Incidentally it took the present author ten years to realise that this was the most promising entrance move.]

2 The discovery of a fascinating kind of symmetry between a, b and z (z defined as a+b+2h) was the next promising step.

3 The next discovery that a, b, z and v have no common factor is crucial.  It is the key early step which leads to the striking result that hp = pabzv, where one and only one of a,b,z,v has the factor p^p-1 times a pth power of a natural number and the other three are pth powers of natural numbers.

4 The discovery that v can be represented by an homogeneous polynomial in a, b, z to the power p was also crucial.

This alone —at a stroke— makes it extremely unlikely that the nominal hypothesis is valid, because the polynomial mentioned above does not look remotely like an expression of the kind p^p-1v^p. It would have to collapse (concertina) progressively downwards term after term into shorter and shorter polynomials until if took the final form of p^p-1v^p or vp.

Any single concertina-type collapse of this kind would be surprising. A set of n-3 sequential collapses of this kind is almost inconceivable.  So here is a wry reflection or spin-off from the main argument which makes it evident that the nominal conjecture (2) is likely to be logically impossible.

The situation which obtains at present is that a considerable body of able people who —if they made the effort— could master the argument in 2 or 3 hours are not bothering to do so, because they think they “know” that “there must be a mistake somewhere”.

This was the feeble excuse given by the Editor and two sets of referees at the Maths Gazette, when they decided in 2020 not to publish it. They should admit that they might be wrong, in which case they are standing in the way of a delightful, valid argument which could give sixth form teachers and learners a much needed lift.  It has been on-line for three years and remains unchallenged.  Anyone who cares about saving the problem-solving reputation of the maths community —which has been content to leave this problem unresolved for more than 350 years— should make an effort to sort this out.

I am offering a prize of £2,000 for the first person to discover a genuinely disabling mistake in the reasoning, i.e. one which plainly invalidates the conclusion. The offer will be open until Christmas Day 2023.

Suggested considered disproofs should be set out in the clearest possible way and sent to the email address: chrisormell@aol.com..

CHRISTOPHER ORMELL 1^st June 2023

If you would like an online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com asking for this.