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 x^{n}+y^{n}=z^{n} 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
The equation
x^{n}+y^{n}=z^{n}
(1)
has for n=3,4,… and natural triples x, y, z no solution
[6,3,4,2]
is usually alleged to be equivalent to Fermat's conjecture about
uniqueness of Pythagoras' Theorem which reads:
It is impossible to split a cube into two cubes, a bi-square into two
bi-squares, and, generally, a power higher than the second into two powers
with the same exponent.
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
l_{x}^{n}+l_{y}^{n}=l_{z}^{n},
(2)
where l_{x}, l_{y}, l_{z} 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
l=rλ,
(3)
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
(xλ)^{n}+(yλ)^{n}=(zλ)^{n},
(4)
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.
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.
FLT is ambiguous because it does not say whether it is meant
to hold for natural triples specifically or for rational or real triples.
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.
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
There is no triple of rational positive numbers x, y, z ∈ Q_{+}
which satisfies Eq.(1) for any exponent n=3,4,….
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 x_{1}, y_{1}, z_{1} which satisfy
Eq.(1) for n=1. Therefore there exist infinitely many
triples of n-dimensional pseudo-cubes with the edge lengths
x_{1}^{1/m}λ, y_{1}^{1/m}λ, z_{1}^{1/m}λ, where
m=1,2,…. So, there exist infinitely many triples
x=x_{1}^{1/m}, y=y_{1}^{1/m}, z=z_{1}^{1/m} for which Eq.(1) is true
if n=m=1,2,….
Due to Pythagoras' Theorem there also exist infinitely many triples
x_{2}, y_{2}, z_{2} which satisfy Eq.(1) for n=2. Therefore there exist
infinitely many triples x=x_{2}^{2/m}, y=y_{2}^{2/m} z=z_{2}^{2/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 (x^{n}+y^{n}) must be equal in size to the number z^{n}, i.e., that
these numbers occupy the same position on the ordered scale of rational
numbers. In other words: The distance of the number (x^{n}+y^{n}) from the
scale's origin must be equal to the distance of z^{n}. So, the analog
criterion is made explicit by the geometrized equation
(x^{n}+y^{n})λ = z^{n}λ.
(5)
The digital criterion is, that the number (x^{n}+y^{n}) 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
x^{n}+y^{n}−z^{n}=F(n, x, y, z)=F(n)
(6)
the condition for (1) being true is compactly expressed by the system
F(n)λ^{n}
=
0,
(7)
F(n)λ
=
0.
(8)
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)λ^{n}=F(r)λ for r=n.
(9)
From (9) there follows according to the rule of Bernoulli/d'Hospital
λ^{n−1}=
lim
r→ n
F(r)
F(r)
=1.
(10)
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.
(11)
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.