The “Difficulties” in Fermat's Original Discourse on the Indecomposability of Powers Greater Than a Square: A Retrospect

The present work is the result of an attempted reconstruction of Fermat’s original discourse along with an explanation of why he might have not written it down. The author had performed it within a one-year period of time – between 1990 and 1993 – trying proving the theorem. When completed, it did look like a proof of Fermat’s epoch, as it only involved the knowledge and techniques available and utilised by Fermat’s contemporary and pre-Fermat mathematical world.


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Fermat's time however, neither algebraic curves nor the notions of 'space', 'transformation', 'groups', etc were known (including to Pierre Fermat; read more on that in Conclusions of this paper) and used to study the properties of natural numbers (and primes).

Proof
The statement of the theorem is rather straightforward and as follows: Neither a cube for two cubes, nor a biquadrate or two biquadrates, and generally no power greater than two can be decomposed into two powers of the same grade. In other words, the equation n n n z y x = + has no solutions in natural numbers, if n is an integer greater than 2.
Therefore, first 1) fix any two arbitrary positive integers m, p such that one of them is greater than the other. Suppose, for example, that m > p and that m and p are coprime integers (i.e. m is not a multiple of p); 2) fix then an arbitrary natural number n. For these three fixed natural numbers: m, p, n, the following that can be expanded or decomposed into a sum according to Newton's binomial (Korn, Zaitsev et al.): Rewrite then (2.6) as n n n (2.7) In order for the y to be a positive integer, n 2 must leave, since for n > 1 n 2 is an irrational number.
It is thus necessary that the expression  i.e. for the case for n = 2 we have also a solution in natural numbers x, y, z. 11) A check showed that for n = 1 or for n = 2 we have solutions of the equation

Conclusion
The "difficulties" were for Fermat the lengthiness of the run of his deductions put in writing, as in the first half of the seventeenth century the mathematical notations had been way far from their present concise and diverse shape, many actions had to be written down in words. Besides, a purely mathematical challenge was that he had to operate the then entirely new notions of binomials and logarithms, both having just appeared for use and to be learnt "on the fly".
Fermat was obviously "playing" with the new notions, decomposing powers of differences into sums of powers and suddenly found out that as one confines oneself with positive integers in the power, the logarithmic equation yields immediately that n n n z y x = + (which is a difference rewritten as a sum) is correct for whole x, y, z only and if only n = 1 or 2.
He (would have) had first to introduce the two new notions so as to fully explain his finding. One can imagine how much room it would take to put down all the deliberations that had led him to his discovery on the margins of a book solely without the proper symbolic notations that a contemporary mathematician avails.
Why Pierre Fermat did not write down all those ideas in a dedicated document is the dedicated question of a dedicated research endeavour. It can come out that he had authored such a separate document indeed, which afterwards was somehow lost oralternativelyhas survived to this day, hidden in an archive or a library or in somebody's unrealised custody.
The author requests the mathematical society to look critically at the deliberations set forth above and to return their assessment.