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(openaccess), 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@opensciencerepository.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
























n2








n1








n
























n+∞








I substitute the
numbers from the rows and columns with generated number C
_{nn}
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
























n2








n1








n
























n+∞








I present the power
of
C_{nn}
as the sum of powers of a pair of numbers
a,
b
or write down this equation in a general form:
С_{nn}^{n}
= a_{nn}
^{n}
+ b_{nn}
^{n}
I emphasize then
that
a_{nn}
^{n}
+ b_{nn}
^{n}
represents
all possible combinations of pairs of the numbers satisfying the
above equation for all possible
C_{nn}
from the matrix. I make a refinement to avoid misunderstandings
assumptions that both numbers are the same for all
C_{nn}
. 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










n2





n1





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 (nth) 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




n2


n1


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. 2730). Springer India.
2. Darmon, H., Diamond, F., & Taylor, R. (1995). Fermat’s last theorem. Current developments in mathematics, 1, 1154.
3. Wiles, A. (1995). Modular elliptic curves and Fermat's last theorem. Annals of Mathematics, 443551.
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, 4764.
Cite this paper
APA
Donev, M. (2013). A Demonstration of the Fermat’s Theorem. Open Science Repository Mathematics, Online(openaccess), e70081937. doi:10.7392/Mathematics.70081937
MLA
Donev, Mitko. “A Demonstration of the Fermat’s Theorem.” Open Science Repository Mathematics Online.openaccess (2013): e70081937. Web. 8 Mar. 2013.
Chicago
Donev, Mitko. “A Demonstration of the Fermat’s Theorem.” Open Science Repository Mathematics Online, no. openaccess (March 8, 2013): e70081937. http://www.opensciencerepository.com/ademonstrationofthefermatstheorem.html.
Harvard
Donev, M., 2013. A Demonstration of the Fermat’s Theorem. Open Science Repository Mathematics, Online(openaccess), p.e70081937. Available at: http://www.opensciencerepository.com/ademonstrationofthefermatstheorem.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.