Open Science Repository Mathematics

doi: 10.7392/Mathematics.70081937


A Demonstration of the Fermat's Theorem


Mitko Donev

Stara Zagora, Bulgaria


Abstract

The thesis of the demonstration is based on the assumption that the equation has at least one correct solution for arbitrary triple positive integers raised to the power n when n > 2.

Keywords: mathematics, Fermat theorem, demonstration.



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

Received: February 7, 2013

Published: March 8, 2013

Copyright: © 2013 Donev, M. Creative Commons Attribution 3.0 Unported License.

Contact: research@open-science-repository.com



Full text


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


The thesis of the demonstration is based on the assumption that the equation has at least one correct solution for arbitrary triple positive integers raised to the power when .


In the demonstration of the theorem, I will use the unique proven equation of this kind and this is the equation of Diofan , known in mathematics as the Pythagorean Theorem. For designation I will construct a matrix of numbers with columns and rows, as the column numbers points to the number in the rows down raised to the power equal to the number of the row of the matrix.


The matrix is presented at this type:




Numbers and columns

1

2

3

4

5

n


Rows and powers

 

 

 

 

 

 

 

1








2








3








4








 

 













 

 













n-2








n-1








n








 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n+∞












I substitute the numbers from the rows and columns with generated number Cnn as the index nn shows the number of the row and number of the columns in when is the integer.


Then the matrix looks like this:




Numbers and columns

1

2

3

4

5

n

n+∞

Rows and powers

 

 

 

 

 

 

 

1








2








3








4








 

 













 

 













n-2








n-1








n








 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n+∞












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


Сnnn = ann n + bnn n


I emphasize then that ann n + bnn n represents all possible combinations of pairs of the numbers satisfying the above equation for all possible Cnn from the matrix. I make a refinement to avoid misunderstandings assumptions that both numbers are the same for all Cnn . I do this for convenience and to avoid writing countless characters, which can lead to confusion and misunderstandings of the thesis.


I show 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+∞

=

=

=

=





And I want to precise that the possible solutions include variants in when the values (read separately) of are rational or negative numbers.


Here's an example as I take the number from column 42, row 3, where:





I present it as the sum from this power of numbers





From this solve is seen that






Where b is a negative number, and the third road of their sum is exactly 42.


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


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


The demonstration will be based on the assumption of opposite assertion, that the equation has a solve for positive integers. It impresses that I don’t mention limits of the range of the power. This is with an aim and I will use it in the demonstration. I will find out all possible solves of the equation in the range of the power n from 1 to plus boundlessness. For this I will use the only proven equation to the moment, and it is at second power, known as Pythagorean Theorem.


It is in interest that Pythagorean Theorem has solves for n pairs of positive integers which are result according to the following from the first three numbers:




3

4

5

6

8

10

12

16

20

…….

…….

…….

N*2

N*2

N*2





Each next row is a result from the previous when doubles its numbers and they satisfy the equality. There are countless solves of the theorem when the numbers are rational or not positive integers.


As the extend of the assertion of Fermat’s theorem excludes the second power of the equation





I look at the theorem of Pythagoras in its whole extend of solves about whole and rational numbers. I will do the demonstration column by column by considering all possible cases.


I start with column 1 equals to one the equations in the column can be solved like a system of equations. I assume that in entire column there is at least one true equation





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


Solve a system of the two true equations and from them I deduce C


;
;


Equating both sides and therefore







To simplify the equation lay two sides as follows






The result is the following equation:





By logarithm find power R on T with which to raise to get K





And substitute K, the result is the following equation





From here the solves of the equation depend on the solve of this equation:





From here the solves of the equations depend on the solve of the following equation


R=2/n


The solves for 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 is so because the number 2 is a prime number and divisible only by 1 and itself, from which it follows that n is a positive integer values only when R = 1 or 2.


Therefore, the solve of the system is possible only when R = 1 or 2 to be true the assertion that the equation has solutions for positive whole numbers.
Analogously construct systems for each row of the matrix and perform the same evidence come to the same conclusion.


;



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


Therefore making similarly systems for each row of the matrix and perform the same evidence comes to the same conclusion.


I find n in the equation were n equals 2 divided by R





The solves of the equations from the first column




Numbers and columns

1

Rows and powers

 

1


2


3


4


 

 

n-2



n-1



n




Depend on the solve f equation for n in the extent (1; 2; 3;………. )


For n in the extent (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 solves of the equation for a positive integer n are two.


I substitute


;


Where R=1 and R=2.


The only possible solves for equals from the first column are


;



According to the Fermat’s theorem these equals are excluded from the extent of the interval for n>2 certain at task condition with which I demonstrate assertion that the other equals have not solves for positive integers.


The same method is used for demonstration about 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 mathematics1, 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.


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