**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.**

**For simple proofs see ****http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.htm**

Proof of Fermat's Last Theorem Using the Exponential Series http://www.coolissues.com/mathematics/Fermat/fermatexp.htm

**http://www.coolissues.com/mathematics/BealFermatPythagoras/proofs.htm**

__http://www.coolissues.com/mathematics/Exponential/series.htm__**
**

**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 series**^{2}**
**

**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 follows**^{3}**:
**

**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 form**^{4}**
**

**in which **** is
Lagrange's (unspecified) number. **

**(4)*** *........* **z*^{m}*
= x*^{m}* +
y*^{m}........* xyz *** 0 **........** ***m***>2**........*x,y,z,m***
****integers**

**and note that the
number ***z*^{m}** 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 ****R**_{n}**
is assumed rational**** R**_{n}**=p/q****
(****p,q**** integers), we can
multiply each side by ****q ****and
obtain ****qR**_{n}**=p****.
This equation is impossible because for large ****n
qR**_{n}** is
a non-vanishing fraction ****<1****
but ****p**** is an
integer ****1****.
The
assumption that ****R**_{n}**=p/q****
is rational is wrong and, since there are only two kinds of real
numbers, rational and irrational, ****R**_{n}**
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=2*^{1/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}

**------------------------------------------------------------------------------------**** **

^{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**