Fermat’s
Last Theorem Revisited
Introduction
Andrew Wiles published a proof of Fermat’s
Last Theorem in The Annals of
Mathematics, 142 (1995). This long work drew upon several arcane branches
of mathematics, most of which are accessible only to professional
mathematicians. The hopes for a simpler proof that could be attributed to
Fermat himself remain unsatisfied. As a consequence of legions of failed
proofs, the use of Pythagorean triples in the solution has been denigrated as the
mark of amateurs, despite the fact that Fermat was engaged in studying them
when he wrote his famous aphorism. If Wiles is right in his opinion that Fermat
did not have a proof of his conjecture then the Pythagorean approach may well
be nugatory. However, a close examination of the relationship between
A^{2 }+ B^{2} = C^{2}
and a^{n} + b^{n} = c^{n}
when n > 2 provides valuable
insights into the nature of the problem.
Fermat’s conjecture can be stated as follows:
There are no positive integers a,b,c,n such that a^{n }+ b^{n} = c^{n }given that^{ }n > 2 and that a,b and c have no common factor. It follows from this last condition that a,b and c must be distinct integers. Formally:
FLT = _{df}
NSabcnK^{2} a^{n} + b^{n} = c^{n} a, b, c, n ∈ ℕ_{1} a,b,c HNCF
It is convenient to specify that a < b,
since this does not affect the argument, and it is easy to show that a and c must be odd integers and b
must be an even integer under the given conditions.
The definition of the conjecture suggests a
strategy for its proof. The contrary of the conjecture NFLT can only be true if
two conditions are jointly satisfied: the equality must be true and all three
variables that a,b and c must be integers. The aim, therefore,
is to show that one of these conditions must be false. Regarding the second condition, it is only
necessary to show that one of the variables
a,b and c must be irrational under the given conditions. It should be noted
that the rational fractional case is subsumed in the integral case.
The key to the Pythagorean approach is the
identity: (a^{n/2})^{2 }=_{ID} ^{ } a^{n}
and the definition
A = a^{n/2},
the consequence being that A^{2 }=_{ID}
a^{n}. The application of this principle to each term of the Fermat
equation produces the identity a^{n}
+ b^{n} = c^{n} =_{ID} A^{2 }+ B^{2} = C^{2}.
The identity symbol signifies more than
the logical equivalence of the two expressions since it also requires that any
rearrangement of one side must be identical to that of the other. For the
purposes of the proof the definitions used are as follows: D_{1}: a^{n}
= A^{2}, b^{n} = B^{2}, c^{n} = C^{2}.
The purpose of deploying the Pythagorean
equation is that the conditions for its truth or falsity can be perfectly
defined. The identity between the equations provides the means of knowing the
truth or falsity of the equality in the Fermat equation. Only the implications
of the truth of a^{n} + b^{n}
= c^{n} are relevant since the falsity of the equation would lead
immediately to the assertion of FLT.
The condition for the truth of A^{2 }+ B^{2} = C^{2}
is given by the following identity:
A^{2 }+
B^{2} = C^{2}
=_{ID}
(x^{2}  y^{2})^{2}
+ 4x^{2}y^{2} = (x^{2} + y^{2})^{2}
under the definitions D_{2}: A = x^{2}
 y^{2}, B = 2xy, C = x^{2} + y^{2}.
Since a^{n}
+ b^{n} = c^{n} =_{ID} A^{2 }+ B^{2} = C^{2} it follows that a^{n} + b^{n} = c^{n} =_{ID }(x^{2}
 y^{2})^{2} + 4x^{2}y^{2} = (x^{2} + y^{2})^{2}.
The consequence is that a^{n} = (x^{2}  y^{2})^{2}, b^{n}
=4x^{2}y^{2} and c^{n}
= (x^{2} + y^{2})^{2}. This provides a model for examining the
implication of a^{n} + b^{n}
= c^{n} a, b, c, n ∈ ℕ_{1} a,b,c HNCF = _{df }NFLT.
The equality b^{n} =4x^{2}y^{2} is the most tractable
for determining the status of the variable
b. Since b^{n} is an even number b
must also be even. The equation (x^{2}
 y^{2})^{2} + 4x^{2}y^{2} = (x^{2} + y^{2})^{2}
is true for all real values of x and y and consequently Pxy (x^{2}  y^{2})^{2} + 4x^{2}y^{2}
= (x^{2} + y^{2})^{2} x,y ∈ ℕ_{1} x,y HNCF
is a theorem.
The expression 4x^{2}y^{2} can only have an nth root when 4x^{2}y^{2 }= 2^{n}u^{n}v^{n},
where u,v HNCF, consequently
b^{n}
= 2^{n}u^{n}v^{n},
and so the definition of b
becomes D_{3}: b = 2uv.
The final definition is D_{4}: v^{n}
= y^{2}, where y and v are odd numbers. These definitions and
key propositions are listed below.
Before proceeding to the formal proof of it is
useful to analyse A^{2 }+ B^{2}
= C^{2} in terms of the derived variables u and v. The rationality
or otherwise of the variables can then be determined.
A^{2}

B^{2}

C^{2}

a^{n}

b^{n}

c^{n}

(x^{2}  y^{2})^{2}

4x^{2}y^{2}

(x^{2} + y^{2})^{2}

(2^{n2}u^{n}  v^{n})^{2}^{}

2^{n}u^{n}v^{n}

(2^{n2}u^{n} + v^{n})^{2}^{}

If 2^{n}u^{n}v^{n} =
4x^{2}y^{2} then u^{n} = 4x^{2}y^{2}
/ 2^{n}v^{n} but v^{n}
= y^{2 }so u^{n} = 4 x^{2} / 2^{n} and u =
x^{2/n}/ 2^{(n2)/n}. It is now clear that u is irrational under the given
condition. This is true whether or not x
is rational. This is relevant because if n
∤
2 then B is
either an irrational square root or is a square. If B =^{ }x^{2}  y^{2} is irrational then b^{n/2
}is an irrational square root unless b is a square.
The significant conclusion is that while a^{n} + b^{n} = c^{n }and
a^{n}, b^{n} , c^{n}
∈ ℕ_{1} the
variable b is irrational.
But if b
∉ ℚ then b ∉ ℕ_{1} . The
irrationality of b is inconsistent
with the condition that b must be a
positive integer as required by the condition on both FLT and NFLT. This
logical contradiction has the form CpNp
, which combined with the theorem CCpNPNp
allows the final conclusion that NFLT is false and so FLT must be true.
The elementary nature of the
mathematics and logic involved suggests that Fermat might well have arrived at
the same conclusion, but neglected to write it down. The use of a variant of
the theorem
CCpCNpp (consequentia mirabilis) could explain
his claim that he had discovered a marvellous proof. The marginalia he could
have squeezed in might then have been as follows:
The supposition a^{n} + b^{n }=
c^{n }, where a, b and c are integers, implies that b is irrational
and so contradicts itself. The contrary
proposition, therefore, must be true, consequentia mirabilis.
Definitions
D_{1}: a^{n} = A^{2}, b^{n}
= B^{2}, c^{n} = C^{2}
D_{2}: A = x^{2}  y^{2},
B = 2xy, C = x^{2} + y^{2}
D_{3}: b = 2uv
D_{4}: v^{n} = y^{2 }
Proposition Condition
T_{1}: SABC A^{2 }+ B^{2} = C^{2}
A,B,C ∈ ℕ_{1} A,B,C HNCF
T_{2}: Pxy (x^{2}  y^{2})^{2}
+ 4x^{2}y^{2} = (x^{2} + y^{2})^{2} x,y ∈ ℕ_{1} x,y HNCF
P_{1}: Sabcn a^{n} + b^{n} = c^{n}
CondP_{1}:
n > 2 a, b, c, n ∈ ℕ_{1} a,b,c HNCF
Formal Proof:
NSabcnK^{2}
a^{n} + b^{n} = c^{n} a, b, c, n ∈ ℕ_{1} a,b,c HNCF
Line

Step

Justification


1

a^{n} =
A^{2}, b^{n} = B^{2}, c^{n} = C^{2 }

D_{1}

df in.

2

A = x^{2}
 y^{2}, B = 2xy, C = x^{2} + y^{2}

D_{2}

df in.

3

b = 2uv

D_{3}

df in.

4

v^{n} =
y^{2}

D_{4}

df in.

5

SABC A^{2 }+
B^{2} = C^{2} (A,B,C Î ℕ_{1 }, A,B,C HNCF)

T_{1}

Th in.

6

Pxy (x^{2}
 y^{2})^{2} + 4x^{2}y^{2} = (x^{2} +
y^{2})^{2} (x,y ∈ ℕ_{1 }, x,y HNCF)

T_{2}

Th in

7

Sabcn
a^{n} + b^{n} = c^{n o } (n > 2, a, b, c, n ∈ ℕ_{1 ,} a,b,c HNCF)

P_{1}

Supposition.

8

b^{n} = 4x^{2}y^{2}

P_{2}

1,2 defs D_{1, }D_{2}

9

b^{n}
= 2^{n}u^{n}v^{n}

P_{3}

1,3 defs D_{1, }D_{3}

10

2^{n}u^{n}v^{n} = 4x^{2}y^{2}

P_{4}

8,9 equality

11

u^{n}
= 4x^{2}y^{2}
/ 2^{n}v^{n}

P_{5}

10 rearrange

12

u^{n}
= 4 x^{2}
/ 2^{n}

P_{6}

11 def 4

13

u
= x^{2/n}/ 2^{(n2)/n}

P_{7}

12 rearrange

14

u
∉ ℚ

P_{8}

13 u is irrational

15

C u ∉ ℚ b ∉ ℚ

P_{9}

14 3, D_{3}

16

b
∉ ℚ

P_{10}

14, 15 MP C exp.

17

C b ∉ ℚ b ∉ ℕ_{1}

P_{11}

16 def
rational

18

b
∉ ℕ_{1}

P_{12}

15.16 MP C exp.

19

C P_{1} b ∈ ℕ_{1}

P_{13}

7, P1 conditions

20

NP_{1}

P_{14}

18, 19 MD C exp.

21

C P_{1}
NP_{1}

P_{15}

7,20 C in.

22

CC P_{1}
NP_{1} NP_{1}

P_{16}

th in. (CCpqq)

23

C P_{1}NP_{1} = P_{15}

P_{17}

21 def in.

24

CP_{15}
NP_{1}

P_{18}

22,23 def P_{15}

25

P_{15}

P_{19}

21 is P_{15}

26

KCP_{15}
NP_{1}P_{15}

P_{20}

24, 25 K in.

27

NP_{1}

P_{21}

26 MP C exp.

28

NSabcn a^{n}
+ b^{n} = c^{n }(n
> 2, a, b, c, n ∈ ℕ_{1 ,} a,b,c HNCF)

P_{22}

7 refuted RA.

Tony Thomas ©
October 2012