James Constant

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

Proof of Fermat's Last Theorem Using the Exponential Series


Fermat (1601-1665) claimed in 1637 to have discovered a marvelous proof of his last theorem.1

(1) ...... 0 ............m>2 ........ x,y,z,m integers

The Binomial Series

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 Bounds of Exponent a

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.

The Remainder Term Rn

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.

Proof and Logic

I will now prove Fermat's Last Theorem by invoking the binomial series (2) and the principle of the excluded middle of traditional logic. This principle holds that either A or not A, where A is Fermat's Last Theorem, equation (1). I will also prove that x,y,z can be arbitrary numbers, positive or negative, rational or irrational. First, I assume not A, i.e., that the following equality holds

(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 whenm=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 0 ........ m>2........ x,y,z,mintegers

which, when expressed as a binomial series, converges or diverges, I show that z cannot be an integer. More specifically, analogous to equation (2):

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

(6) y<x

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, sincex 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/mis 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)


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 setsx = 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

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