Andrew Wiles solved the master problem whether Fermat’s conjecture was valid more than 25 years ago. He used highly-abstract modern concepts which were miles away from anything Fermat could have known. It was a triumph in a category (i.e. resolving notorious uncertainties) which has been conspicuously neglected for a long time. It also, incidentally, increased the credibility of Fermat’s claim that he had a valid (elementary) solution in the 17^th century. An elementary solution would be a cause for much rejoicing, because many thousands of mathematically literate people could participate in the experience, not merely a handful of hyper-specialists. But a strange wall of resistance still persists facing any attempt to find Fermat’s lost reasoning.  This is obfuscation. We urgently need to find Fermat’s lost reasoning today, to restore faith in the power of reasoning in elementary maths. This Blog (26) completes the elementary re-run of the classic argument set out in Blogs 1 and 2. 

The author’s Elementary Fermat argument  This was published on line in Blogs (1) and (2) in 2020 after being turned down by the Mathematical Gazette on the grounds that although the referees could not find a mistake, <<there must be a mistake in it somewhere>>!  What does this brazenly presumptive dismissal say? It says that there is a professional animus against any attempt to rediscover Fermat’s lost elementary proof. This has become, in effect, a taboo: one which is perpetuating obfuscation over enlightenment, and ensuring that nothing will ever be done to resolve the situation.  It is dooming the elementary subject to a deep subliminal sense of inadequacy, because Pierre de Fermat, a capable mathematician, probably had something essentially similar (i.e. a really elementary) proof in his hands more than 350 years ago! It is a scandal of the first order that capable higher mathematicians have left this situation unresolved for all this time. They should have appointed a task force in the 17^th century with the distinct instruction to <<put-in whatever effort is necessary to rediscover the missing proof>>.  

   A number of friends and acquaintances (accomplished higher mathematicians) have taken a quick look at my 2020 argument, but they have made it clear that they are determined at all costs to “find” instant, trivial, imagined “objections”. This has then provided them with a flimsy excuse which —they have supposed— justifies them in abandoning the enquiry. 

   In response, the 24th blog on mathsforrenewingreason.com shows plainly how the proof applies for fifth powers (x⁵ + y⁵ = z⁵), in the major case when the inner tray T_ab of (a+b)⁵ is not divisible by 5.  After dividing the inner tray by 5 one could get a remainder of 4,3,2,1 or 0: so this condition covers 4/5ths of the possibilities. So the form of proof offered by the earlier posts covers 80% of the cases when p=5. For larger prime numbers the valid percentage gets larger and larger: for example, when p= 113 the probability that T_ab is not divisible by 113 is 99.115%. We know that most of the infinite set of prime numbers are larger than the greatest known prime which is currently 2^82589933 -1, a number p_m, with nearly 25 million digits.   So the inner tray T_ab is equally likely not to be divisible by p_m  in  2^82589933  – 2 of the cases, an unbelievably overwhelming majority. Now an infinite host of prime numbers are larger than p_m, (compared with a finite number less) so the norm is that an unbelievably vast majority of residues are > 0 and the proven case covers 99.9999…and so on… % of the possibilities, the ‘and so on’ here signalling about 25 million decimal 9s. 

   When p = 5 the majority case argument is much easier to follow than that of the original 2020 generalised proof. It can also easily be seen to generalise to whatever value we substitute for 5. It will, no doubt, finally become recognised, but don’t hold your breath, because confidence in maths at all levels is currently at an all-time low. 

The ‘Glitch’ mentioned in Blog 25 

In the previous Blog (25) a “glitch” was reported in the original 2020 minor argument. However closer inspection shows that it was only the relatively slight matter that Y and L were mistreated as natural numbers. Proposition (37) was that when p =- 7

(Y+N₁ABZ)⁷/7L⁷ – b⁴ should have the factor A.  …………………………..(1)     (This builds-in the requirement that a cascade of merging terms can occur.)

We can find out whether the requirement (1) is met by substituting A=0 in (1).

When A=0 this becomes Y⁷/7L⁷ – (B⁴)⁷   …………………………………………(2)

But when A= 0, L and Y become specific numbers which are denoted L and Y;

they are not, however, necessarily natural numbers or even rational numbers. 

We look at the specifics here:   [7⁶ = 117649.]

When A=0, T_ab = b⁴. And v = 117649V⁷ = T_ab, so V = (b⁴/117649)^1/7. 

Now Y = T_ab/7V, or b⁴/ b^4/7/7^13/7 or b^24/7/7^13/7 .

L = T_ab/49V² . or b⁴/49(b⁴/117649)^2/7 or b^20/7/7^2/7  .

So (2) becomes (b²⁴/7¹³)/(b²⁰/7) – b⁴, or b⁴/7¹² – b⁴  which is clearly not zero.

It follows that the condition (1) is not met and that this is a contradiction, because the expression for v is composed entirely of terms with the factor A apart from b⁴ at the far end of T_ab.

   Substituting any other prime number for ‘7’ in this reasoning obviously generalises the argument to all possible cases except p=2 and powers of 2. 

   What is absolutely clear is that v-b⁴ has the factor A and that putting A=0 will turn it into zero. But if we build into v the precondition that the cascade of merging can occur, it no longer equals zero. This shows quite emphatically that the pre-condition can’t be met.

   This establishes the minor case of the Fermat Result when p=7. It is easier to follow in this specific case rather than the general case. p=7 is the simplest minor case, because when p=5, the minor case does not arise. 

   Both the major and minor cases have now been established when p= 5 and p = 7 respectively, but of course this is just a clear sign that the form of reasoning in the general case is valid. It should be a moment for much rejoicing, because a dark cloud which has hung over elementary mathematics since the early 17^th century, has, at last, been resolved. However morale in both elementary and higher maths is currently at an all-time low, and it is anyone’s guess how long it will take for this relief to be widely felt.

CHRISTOPHER ORMELL 1^st July 2022