Open Science Repository Mathematics

doi: 10.7392/Mathematics.70081937

A Demonstration of the Fermat's Theorem

Mitko Donev

Stara Zagora, Bulgaria

Abstract

This demonstration rests on the premise that the equation admits at least one valid solution for any triple of positive integers raised to the power n, where n > 2.

Keywords: mathematics, Fermat theorem, demonstration.

Full Text

The equation has not determination for positive integers with degree when .


The argument of this demonstration assumes that the equation possesses at least one valid solution for any triple of positive integers raised to the power when .


For this demonstration of the theorem, I rely on the only proven equation of this type, namely the Diophantine equation , known in mathematics as the Pythagorean Theorem. For notation, I will build a matrix of numbers with columns and rows, where the column numbers indicate the number in the rows below raised to the power equal to the row number of the matrix.


The matrix is shown in this form:

Numbers and columns	1	2	3	4	5	n
Rows and powers
1
2
3
4
n-2
n-1
n
n+∞

I replace the numbers from the rows and columns with a generated number C_nn, where the index nn indicates the row number and column number when it is an integer.


The matrix then appears as follows:

Numbers and columns	1	2	3	4	5	n	n+∞
Rows and powers
1
2
3
4
n-2
n-1
n
n+∞

I express the power of C_nn as the sum of powers of a pair of numbers a, b, or write this equation in a general form:


С_nnⁿ = a_nn ⁿ + b_nn ⁿ


I stress then that a_nn ⁿ + b_nn ⁿ stands for all possible pairs of numbers satisfying the above equation for every possible C_nn from the matrix. I add a clarification to avoid the misunderstanding that both numbers are the same for all C_nn. I do this for convenience and to avoid writing countless characters, which could lead to confusion and misunderstanding of the thesis.


I set out the order of all possible equations in the matrix.

Numbers and columns	1	2	n	n+∞
Rows and powers
1
2
3
4
n-2
n-1
n
n+∞	=	=	=	=

I also want to clarify that the possible solutions include cases where the values (taken separately) of are rational or negative numbers.


Here is an example taking the number from column 42, row 3, where:





I express it as the sum of this power of numbers





From this solution it is seen that






Here b is a negative number, and the third root of their sum is exactly 42.


Let, in the matrix described above, the row and columns summarize equations in all cases of such solutions. These solutions satisfy the thesis of the theorem and are not of interest to the demonstration. They meet the condition that the equation has no solutions for positive integers.


Let the (n-th) row and column of the matrix summarize all possible cases for solving the equations, when the numbers a, b, c and their power n are positive integers.


The demonstration will rest on the assumption of the opposite claim, that the equation has a solution for positive integers. It is notable that I do not mention limits on the range of the power. This is intentional and I will use it in the demonstration. I will determine all possible solutions of the equation over the range of the power n from 1 to plus infinity. For this I will use the only proven equation to date, which is at second power, known as the Pythagorean Theorem.


It is interesting that the Pythagorean Theorem has solutions for n pairs of positive integers that result from the first three numbers as follows:

3	4	5
6	8	10
12	16	20
…….	…….	…….
N*2	N*2	N*2

Each successive row results from the previous one when its numbers are doubled, and they satisfy the equality. There are countless solutions of the theorem when the numbers are rational or not positive integers.


Since the scope of the assertion of Fermat’s theorem excludes the second power of the equation





I consider the theorem of Pythagoras over its full range of solutions for whole and rational numbers. I will carry out the demonstration column by column, considering all possible cases.


I begin with column 1, equal to one; the equations in the column can be solved as a system of equations. I assume that in the entire column there is at least one true equation





As proven by Pythagoras in the second row of the matrix:


Solving a system of the two true equations, I deduce C from them


;
;


Equating both sides, and therefore







To simplify the equation, set the two sides as follows






The result is the equation below:





Using logarithms, find the power R of T to which it must be raised to obtain K





Substituting K, the result is the following equation





Hence the solutions of the equation depend on the solution of this equation:





Hence the solutions of the equations depend on the solution of the following equation


R=2/n


The solutions of the equation for n (1,2,3,4……….. ) is:






…………..
or


For the equation to be true





And, if there are countless solves for positive integers, than in the interval for its





It is necessary to discover the countless positive integers, but because of the range consists of fractions of which the dividend is the number 2, the only positive whole numbers are 1 and 2. This holds because the number 2 is prime and divisible only by 1 and itself, from which it follows that n is a positive integer only when R = 1 or 2.


Therefore, the system can be solved only when R = 1 or 2 for the assertion that the equation has solutions for positive whole numbers to hold.
Analogously, constructing systems for each row of the matrix and performing the same proof leads to the same conclusion.


;



The result is that n is a positive integer only when R equals R=1 or R=2.


Therefore, constructing similar systems for each row of the matrix and performing the same proof leads to the same conclusion.


I find n in the equation, where n equals 2 divided by R





The solutions of the equations from the first column

Numbers and columns	1
Rows and powers
1
2
3
4
n-2
n-1
n

Depending on the solution of the equation for n in the range (1; 2; 3;………. )


For n in the range (1; 2; 3;………. ) all possible equations are:

	n=	Where R=1	Where R=2
powers		n=‌	n=‌
1
2
3
4
……	…………	………..	……….
n

Where R=1 and R=2


Therefore the solutions of the equation for a positive integer n are two.


I substitute


;


Where R=1 and R=2.


The only possible solutions for the equalities from the first column are


;



According to Fermat’s theorem, these equalities are excluded from the scope of the interval for n>2 set by the task condition, with which I demonstrate the assertion that the other equalities have no solutions for positive integers.


The same method is used to demonstrate the result for all other columns of the matrix.

Connections

1. Alladi, K. (2013). Fermat and Ramanujan: A Comparison. In Ramanujan's Place in the World of Mathematics (pp. 27-30). Springer India.

2. Darmon, H., Diamond, F., & Taylor, R. (1995). Fermat’s last theorem. Current developments in mathematics, 1, 1-154.

3. Wiles, A. (1995). Modular elliptic curves and Fermat's last theorem. Annals of Mathematics, 443-551.

4. Stewart, I., & Tall, D. O. (2002). Algebraic number theory and Fermat's last theorem. AK Peters.

5. Kleiner, I. (2012). Fermat’s Last Theorem: From Fermat to Wiles. Excursions in the History of Mathematics, 47-64.

Cite this paper

APA

Donev, M. (2013). A Demonstration of the Fermat’s Theorem. Open Science Repository Mathematics, Online(open-access), e70081937. doi:10.7392/Mathematics.70081937

MLA

Donev, Mitko. “A Demonstration of the Fermat’s Theorem.” Open Science Repository Mathematics Online.open-access (2013): e70081937. Web. 8 Mar. 2013.

Chicago

Donev, Mitko. “A Demonstration of the Fermat’s Theorem.” Open Science Repository Mathematics Online, no. open-access (March 8, 2013): e70081937. http://www.open-science-repository.com/a-demonstration-of-the-fermats-theorem.html.

Harvard

Donev, M., 2013. A Demonstration of the Fermat’s Theorem. Open Science Repository Mathematics, Online(open-access), p.e70081937. Available at: http://www.open-science-repository.com/a-demonstration-of-the-fermats-theorem.html.

Science

1. M. Donev, A Demonstration of the Fermat’s Theorem, Open Science Repository Mathematics Online, e70081937 (2013).

Nature

1. Donev, M. A Demonstration of the Fermat’s Theorem. Open Science Repository Mathematics Online, e70081937 (2013).

doi

Research registered in the DOI resolution system as: 10.7392/Mathematics.70081937.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.

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