The higher maths establishment seem to insist, in effect, that no elementary version of Fermat’s lost theorem will, or could, ever be found. I think this is the best interpretation one can attach to their taboo against any notion that the problem needs to be re-opened. History appears to support their case, because more 350 years have passed since Fermat died, and no one has managed to reconstruct his missing proof —apart from the present author whose 2019 proof was turned down by The Mathematical Gazette after two sets of referees and the Editor himself had failed to find a mistake. They were, however, able to field the effective presumptive argument <<That there must be a mistake in it somewhere!>>.

    Kant famously argued that no principle of moral judgment could be valid unless it was capable of being applied universally. This was the basis of his Categorical Imperative. If the referees and Editor at The Mathematical Gazette were to apply their form of effective presumptive rejection to all the articles submitted to their journal, they would end-up with nothing to print! 

I think this shows that the effective presumptive rejection principle may contain a flaw of some kind.

This is a minor setback, though, because the argument is on-line, and had now been strengthened by posts 24 and 26 of this blog. There are many able, independent-minded mathematicians out there with more-than-average curiosity. It is likely that one of them will overcome her/his initial reluctance to devote time to carefully following the elementary sequence of reasoning when so much hinges on the result. Thy will find that the new approach to Fermat is sound, logical and very interesting.  (To “bring this out” was the purpose of posts 24 and 26 in this series: they set out the form of reasoning involved when p was 5 or 7.  It is much easier to see that the argument is cogent and valid when dealing with such specific cases.) 

     The dearth of previous serious elementary attacks on the Fermat problem, is, though, a scandal of the greatest magnitude.  How can any mathematician simply happily write-off the agonising loss of Fermat’s original reasoning as <<just one of those things>>?  To think that this is not a cause for great regret, is to rubbish Fermat’s amazing achievement when he solved a problem which has proved too difficult for anyone to replicate in 350 years.

     I’m afraid the default inference here is all-too plain: that the higher maths establishment don’t want an elementary solution to be found.  Partly, no doubt, this is down to their desire to protect Andrew Wiles’ historic achievement —in breaking through spectacularly and solving the problem in 1995. But there is also the niggling additional thought that a clear-cut elementary solution might offer lucidity which would  upstage the much more arcane abstractions involved in Wiles’ proof.

Ever since Gauss proved the result for p=3 —using complex numbers— there has been a tacit, unspoken tendency for higher mathematicians to think that their performance skills can only be improved by adopting more and more attenuated generalisations. Excelsior!  Ever upwards! But the most startling bits of powerful pure mathematics ever discovered were already known at the time of Euclid (c. 300 B.C.E).  They only involve the simplest concepts.  The lucidity of these old results is very special.

We seem to have lost the confidence in our ability to attack mathematic problems by the simple dog-worrying-a-bone method… involving a lot of pausing, thinking and total familiarisation around the frustrating conceptual blind-spot which is barring progress.

The present author decided to attack the problem of Fermat’s lost theorem in January 1991. His idea was to devote about an hour a day to the project. He had one special advantage viz. that he had no prior knowledge whatever of modern number theory. 

Prior knowledge isn’t going to help, because it is inevitably coloured by the Excelsior, Ever Upwards modern maths bias of the last 300 years. By contrast the present author conceptualised his ambitious problem-solving project as a ‘bare hands’ struggle, which would probably involve digging up some unknown ,  new, unfamiliar —but very elementary— concepts of the kind we teach to 17 year olds.  (It did.)

His project took far longer than he ever envisaged at the beginning.  On about 60 occasions he had the flashing idea that he was “there”… only to discover later that it was a false dawn. It took 29 years and more than 10,000 hours in total.  This will turn out to have been a waste of time, if laziness prevails and no one can be bothered to enjoy this delightful, unexpected narrative.

CHRISTOPHER ORMELL 1^st August 2022