Issues about Fermat's "Last Theorem"

Ernst Terhardt, Tech. Univ. München, Germany

May 1 2022


Fermat's so-called Last Theorem is a bad conjecture; it is ambiguous and misleeading; it is not equivalent to Fermat's original conjecture which proposes that Pythagoras' Theorem is unique. The two conjectures need to be revised into the proposition that the equation xn+yn=zn can for n=3,4,… not be satisfied by rational triples x, y, z ∈ Q+. An elementary proof of the unified conjecture is described.

1  Introduction

1.1  Two different conjectures

Fermat's so-called Last Theorem, FLT, i.e., the proposition is usually alleged to be equivalent to Fermat's conjecture about uniqueness of Pythagoras' Theorem which reads: This conjecture shall be labeled Fermat's original conjecture, FOC.

1.2  Geometric and analytic aspects of FOC

Since Pythagoras' Theorem, PYT, is a geometric proposition, FOC needs to be consistent with the proposition that the equation
where lx, ly, lz are the edge lengths of n-dimensional pseudo-cubes, may be satisfied only for n=1 and n=2. With the definition of geometric length, distance or coordinate
where l is length, r is real positive and λ is a place holder for an arbitrary unit of length, Eq.(2) gets expressed by
where x, y, z are real positive length factors. FOC in effect proposes that Eq.(1) can for n=3,4,… not be satisfied by any triple of real positive length factors x, y, z ∈ R+.
If FOC is correct then FLT is correct also, because the natural numbers are a subset of the real numbers.

1.3  Issues

FLT is affected by the following issues.
  1. In contrast to a prevalent presumption, FLT is not an equivalent rendition of FOC. FLT's focus on natural triples is not a part of FOC.
  2. FLT is ambiguous because it does not say whether it is meant to hold for natural triples specifically or for rational or real triples.
  3. FLT ismisleading because it implicitly suggests that it is meant to hold for natural triples specifically. The world's mathematics enthusiasts have collectivly fallen into this trap and faithfully take it for granted that FLT poses a problem of integer-number theory.
  4. A proof of FLT, i.e., of insolubility of (1) for n=3,4,… and natural triples, does not per se answer the question whether FOC holds for real or rational triples x, y, z or for natural triples spedcifically. The proof of FLT by Wiles [7] and Taylor [5] gives no explicit answer to that question.

1.4  Outlook

As it turns out, insolubility of (1) for n=3,4,… can not be specific to natural triples because insolubility for natural triples inevitably goes along with insolubility for non-integer rational triples. On the other hand, Eq.(1) turns out to have solutions for n=3,4,… and certain irrational triples. Both FLT and FOC need to be revised accordingly.
The geometric implications of Eq.(1) lead to a concise elementary proof of Fermat's conjecture.

2  Revision of FLT

Insolubility of (1) for some exponent n is only specific to natural triples x, y, z if it can not go along with insolubility for some non-integer rational or some irrational triple and the same exponent.
Proposition. Insolubility of (1) for natural triples invariably goes along with insolubility for non-integer rational triples x=p/q, y=r/s, z=t/u; p, q, r, s, t, u ∈ N.
Proof. If it is tentativly yassumed that (1) is true for some non-integer rational triple then multiplication of (1) by (qsu)n on both sides turns the equation into an expression for the natural triple psu, rqu, tqs, which is true also. This however is incompatible with insolubility of (1) for natural triples. So, if (1) is insoluble for natural triples then the equation can not be soluble for any non-integer rational triple and the same exponent. Equation (1) either is insoluble for rational triples or not at all.
Conclusion. FLT needs to be revised into RFLT, i.e., into the proposition

3  Revision of FOC

FOC differs from the revised version RFLT of FLT by FOC's proposition that (1) is insoluble for n=3,4,… and real positive triples x, y, z. As it turns out, FOC needs to be revised also, i.e., to be restricted to rational triples, because there exist certain irrational triples for which (1) may be soluble for any exponent n=3,4,….
There evidently exist infinitely many triples x1, y1, z1 which satisfy Eq.(1) for n=1. Therefore there exist infinitely many triples of n-dimensional pseudo-cubes with the edge lengths x11/mλ, y11/mλ, z11/mλ, where m=1,2,…. So, there exist infinitely many triples x=x11/m, y=y11/m, z=z11/m for which Eq.(1) is true if n=m=1,2,….
Due to Pythagoras' Theorem there also exist infinitely many triples x2, y2, z2 which satisfy Eq.(1) for n=2. Therefore there exist infinitely many triples x=x22/m, y=y22/m z=z22/m for which (1) is true if n=m=1,2,….
The triples of the first class are for m=2,3,… in general irrational and the triples of the second class are for m=3,4,… in general irrational.
Conclusion. Since there exist irrational triples x, y, z which satisfy (1) for n=3,4,…, FOC needs to be restricted to rational triples x, y, z. The revised version RFOC of FOC is identical to RFLT.

4  Proof of RFLT and RFOC

For the expression (1) to be true, an analog and a digital condition must both be satisfied. The analog condition is, that the rational number (xn+yn) must be equal in size to the number zn, i.e., that these numbers occupy the same position on the ordered scale of rational numbers. In other words: The distance of the number (xn+yn) from the scale's origin must be equal to the distance of zn. So, the analog criterion is made explicit by the geometrized equation
(xn+yn)λ = znλ.
The digital criterion is, that the number (xn+yn) must be a power with the exponent n. This criterion becomes geometrically explicit when in (1) the numbers x, y, z are converted into lengths, i.e., by attachment of the unit λ to them. This variant of geometrization of (1) yields the equation (4).
With the definition
xn+yn−zn=F(n, x, y, z)=F(n)
the condition for (1) being true is compactly expressed by the system
The proof of RFLT and RFOC emerges from testing consistency of the system (7, 8) with the tentative assumption that (1) is true, i.e., that F(n)=0. To that purpose the exponent n is conceptualized as the solution exponent of the equation F(r)=0, where r ≥ 0 is a continuous real variable. So, with the tentative assumption F(n)=0 Eqs.(7, 8) become expressed by the criterion
F(r)λn=F(r)λ  for r=n.
From (9) there follows according to the rule of Bernoulli/d'Hospital
r→ n 

The solution of Eq.(10) is n=1. So, where Eqs.(4, 5, 7, 8) are concerned, Eq.(1) is insoluble for n ≠ 1 and rational triples x, y, z. This result formally corroborates the propositions of FLT and FOC to be true but is in conflict with Pythagoras' Theorem, i.e., with solubility of (1) for n=2.
This conflict is overcome by involvement of an additional equation, namely, the definition (3) of geometric length. This definition warrants Eq.(9) to be true for n=2, because the criterion (9) reads for n=2,
F(2)λ = 0  for  F(2)=0.
Conclusion. RFLT and RFOC, i.e., the revised versions of FLT and FOC, are correct in proposing that there is no triple of rational numbers x, y, z ∈ Q+ which satisfies (1) for some exponent n=3,4,….

5  Discussion

With respect to the proof of FLT by Wiles [7] and Taylor [5] it is remarkable that FLT and FOC are by revision into RFLT, RFOC unified, holding for rational triples x, y, z. The Wiles/Taylor proof is based on the theory of modular elliptic curves, which are defined for rational triples x, y, z. So, that proof in effect corroborates insolubility of (1) for n=3,4,… and natural triples x, y, z by proving insolubility for rational triples. The Wiles/Taylor proof and the proof described in Sec.4 are consistent with one another.
The wording of Fermat's original conjecture reveals that Fermat, although he was engaged in integer-number theory, did not take the problem of uniqueness of Pythagoras' Theorem one-sidedly for a number-theoretic problem. Fermat was an early contributor to René Descartes' concept which today is called analytic geometry [1]. It may be surrmised that Fermat was aware of the geometric implications of Eq.(1) and that he was aware of the fact pointed out in Sec.2, that FLT can not be specific to natural numbers. As is well known, Fermat was reluctant about communicating explanations abbout his mathematical discoveries [6,4].

Acknowledgment Michael Terhardt patiently read many drafts of this article and provided helpful comments.


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