...........xyz 
0 ............m>2 ........ x,y,z,m integers PROOF OF FERMAT'S LAST THEOREM
James Constant
math@coolissues.com
Fermat's Last Theorem is solved using the binomial series. A new theorem determining the irrationality of a number using its infinite series expansion is presented.
Introduction
Fermat (1601-1665) claimed in 1637 to have discovered a marvelous proof of his last theorem.1
(1) ...... ...........xyz 0 ............m>2 ........ x,y,z,m integers
Before seeking the solution to Fermat's last theorem, consider the binomial series2
in which a is an arbitrary number, positive or negative, rational or irrational.
The exact conditions under which the series, equation (2), is convergent are as follows3:
1. If index a
is an integer 
the series terminates
and is valid for all values of x
and becomes the Binomial Theorem.
2. For all other
values of a, the
series is absolutely convergent for 
and
divergent for 
.
3. For x=+1
the series converges absolutely if a>0,
converges conditionally if -1<a<0,
and diverges if a
Finally, at x=-1
the series is absolutely convergent if a>0,
divergent if a<0.
Equation (2) can be written as the sum of its first n terms and a remainder function, in Lagrange's form4
in which 
is
Lagrange's (unspecified) number.
(4) ........ zm
= xm +
ym........ xyz 0 ........ m>2
........
x,y,z,m integers
and note that the number zm is an integer if z is so. I also note that equation (4) fails when m=0 but succeeds when m=1 (addition of numbers) or m=2 (Pythagorean theorem). For all other numbers ,
(5) ........
z = x (1 +
(
)m)1/m ........
xyz 
0 ........ m>2
........ x,y,z,m
integers
1. Since m>2,
index 1/m is not an integer 
and the
series cannot terminate becoming
the Binomial Theorem.
2. The
series is absolutely convergent for 
and
divergent for 
.
It is easy to show that the parenthesis term in equation (5) is an irrational number, since
from which we
immediately see that 
.
Equation (6) says that the left hand
parenthesis term equals a rational number plus an irrational
non-vanishing fraction since, if Rn
is assumed rational Rn=p/q
(p,q integers), we can
multiply each side by q and
obtain qRn=p.
This equation is impossible because for large n
qRn is
a non-vanishing fraction <1
but p is an
integer 
1. The
assumption that Rn=p/q
is rational is wrong and, since there are only two kinds of real
numbers, rational and irrational, Rn
is irrational. The right hand sum of equation (6) is, therefore,
an irrational number. Accordingly, the fraction z/x
obtained from equation (5) cannot be a rational number and, since
x is an integer, z
is irrational and cannot be an integer when 
.
3. For y/x=+1 the series converges absolutely since m>2 and thus 1/m>0. The same argument applies. In this case z=21/m is a non-integer.
Therefore, the assumed equality of equation (4) is wrong for the stated conditions x,y,z,m integers and, by the principle of the excluded middle of traditional logic, Fermat's Last Theorem, equation (1), is proven. However, the assumed equality of equation (4) is correct for other values of x,y,z,m. In other words, the binomial series (2) confirms Fermat's inequality for the stated conditions x,y,z,m integers but discomfirms Fermat's inequality for other values of x,y,z,m.
New Theorem
If a number 
can be
represented by a converging infinite series whose first n terms 
is a rational number and whose remainder 
is a non-vanishing fraction 
then the number 
is an irrational
number. For
if the contrary is true,i.e. if 
is
rational (p,q integers),
then since 
is a rational fraction (p',q'
integers)
(7) 
which says that a rational fraction p/q,
equals a rational fraction p'/q'
plus a non-vanishing fraction 
, which is
impossible. Formally, if we solve equation (7) for 
and multiply each side of the resulting equation
with the integer qq' we
obtain
(8)
......
which says that an integer 
equals a non-vanishing fraction
. Equation (8) is impossible because
qq' is a fixed integer and thus 
as n increases.
Pythagoras, Newton, and Fermat
The question why z
is a real (rational or irrational) number when m=2,
but is an irrational number when m>2
is beyond this paper. It is an issue between Pythagoras
(geometry) and Newton (binomial series). For example, if one sets
x = x'(1
+ (
)m)1/m
in equation (5), where x'
is an integer, the resulting series for z
terminates and the equality in equation (5) holds for m=1
and m=2. In other words,
the binomial series (2) confirms the addition of numbers when m=1
and Pythagora's theorem when m=2
for x and
z absolutely converging infinite series and y
any arbitrary number. These conclusions fall short of the known
universal applicability of the addition of numbers and
Pythagoras' theorem for arbitrary numbers x,y,z. The question
remains why the binomial series does not fully explain the
addition of numbers and Pythagoras' theorem. Nor can geometry
prove Pythagoras' theorem when at least x
or y is an absolutely
converging infinite sum. Perhaps geometry and the binomial series
are complementary; the former contained in the domain of real
numbers the latter in the domain of absolutely converging
infinite sums.
The binomial series used in the present proof, equation (2), was invented by Newton (1642-1772) about 1676.5 Fermat's claimed proof of equation (1) could not, therefore, have been based on the results of equations (2) through (4). A recent book gives a detailed history of Fermat's Last Theorem and attempts to solve it.6
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1 E. Bell, The Development of Mathematics, Dover Publications, Inc., New York 1972 page 157.
2 e.g. see R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York 1947 Vol I pages 329-330.
3 R. Courant note 2 above page 406.
4 R. Courant note 2 above page 337.
5 E. Bell note 1 above page 406.
6 Simon Singh, Fermat's Last Theorem, Fourth Estate Ltd, 6 Salem Road, London W2 4BU,England.
Copyright 
2003 by James Constant
By same author: http://www.coolissues.com/mathematics/sameauthor.htm
