The Proof of the Fermar''s last theorem
Liao Teng
Tianzheng International Mathematical Research Institute, Xiamen, China
Abstract:
In order to strictly prove a conjecture about the solution of positive integers of indefinite
equations proposed by French scholar Ferma around 1637 (usually called Ferma''s last theorem)
from the perspective of pure mathematics, this paper uses the general solution principle of
functional equations and the idea of symmetric substitution, as well as the inverse method. It
proves that Fermar''s last theorem is completely correct.
Key words:
Fermat indefinite equation, functional equation decomposition, symmetric substitution, prime number
principle, proof by contradiction.
I. Introduction
Around 1637, the French scholar Fermat, while reading the Latin translation of Diophatus''
Arithmetics, wrote next to proposition 8 of Book 11: "It is impossible to divide a cubic number into the
sum of two cubic numbers, or a fourth power into the sum of two fourth powers, or in general to divide
a power higher than the second into the sum of two powers of the same power. I am sure I have found a
wonderful proof of this, but the blank space here is too small to write."
Around 1637, the French scholar Fermat, while reading the Latin translation of Diophatus''
Arithmetics, wrote next to proposition 8 of Book 11: "It is impossible to divide a cubic number into the
sum of two cubic numbers, or a fourth power into the sum of two fourth powers, or in general to divide
a power higher than the second into the sum of two powers of the same power. I am sure I have found a
wonderful proof of this, but the blank space here is too small to write."
Since Fermat did not write down the proof, and his other conjectures contributed a lot to
mathematics, many mathematicians were interested in this conjecture. The work of mathematicians has
enriched the content of number theory, involved many mathematical means, and promoted the
development of number theory.
II. Reasoning
n n n
Ferma''s last theorem says that equation x +y =z (x ， ，x
n n n
) has no positive integer solution, if x +y =z (x ， ，x
) is true,
n n p-k-h(i) p-k-g(j) p-k-h(i) p-k-g(j) p-k-h(i) p-k-g(j) n n
then =x + u v u v +… u v + …+y >z (x
n n n
), so If x +y =z (x ,x )
n n
has a positive integer solution, then x+y z, and x+y>z. So if =z (x ， ，
n n n
x ， ) is true ，then x +y =z (x ， ，x
p p p
) has no positive integer solution.When x +y =z (x p is any prime
p p p p p p
number, p 3)is true , then because 2 x +2 y =2 z
(x ,p 3), when if 2x=u+v (x ， ，
2y=u-v (x ， , then let''s put
2x=u+v(x ， ， and 2y=u-v (x ， ，
p p p p p p
into 2 x +2 y =2 z (x ),So(2u)
p-1 p-3 p-3 p-4 p-4 p-k-h(i) p-k-g(j) p-1
(u u v + u v +… + u v +… + pv )
p-1
=(2z)(2z) ( k
， ， ,
p-1 p-3 p-3 p-4 p-4 p-k-h(i) p-k-g(j) p-1 p-1
then (u)( u u v + u v +… + u v +… + pv )=(z)(2z)
p-1 p-1
=(z)(2) (z) ( k
， ,
Since u is a positive integer product factor of the value on the right-hand side of the
equation,
p p p
When 2x=u+v +v, then x> , then x +y >z
p p p
( x x ),then x +y =z
( x x ) has no positive integer
solution. So let''s just think about u=z and .When
p-1 p-3 p-3 p-4 p-4 p-k-h(i) p-k-g(j) p-1 p-1 p-1
u=z,then(u u v + u v +… + u v +… +pv (2) (z)
( k
. And 2 = + then u=x+y, according to u z, then
p p p p p
x+y z . When x+y z, then z , then x +y p p p
,x ， ) , so x +y =z
( k
. And 2 = + then u=x+y, according to u z, then
p p p p p
x+y z . When x+y z, then z , then x +y p p p
,x ， ) , so x +y =z
( x x ) has no positive integer
p p p
solution, this contradicts the previous assumption that x +y =z (x ≠y ≠z ≠1,p is any prime
number,p ≥3) has positive integer solutions, and when then according to u=x+y, then
p p p p p
x+y z ,and when x+y z, then z , then x +y ,x ， ) ,so
p p p
x +y =z ( x x ) has no positive
p p p
integersolution, this contradicts the previous assumption that x +y =z (x ≠y ≠z ≠1,p is any prime
number,p ≥3) has positive integer solutions.Therefore, it is wrong to assume that if p is a prime
p p p
number, then x +y =z (x ≠y ≠z ≠1,p is any prime number,p ≥3) has a positive integer solution ，so
for any prime number
p p p
p,x +y =z (x ,x ) has no
n n n
positive integer solutions. So the fermat equation x +y =z (x , x
) has no positive integer solutions.
III.Conclusion
Fermar''s last theorem is absolutely correct.
IV.Significance and prospect
After the solution of Ferma''s last theorem conjecture, it is of great significance to the development of
number theory, and readers can do something in the field of stacked number theory.
V.Thanks
Thank you for reading this paper.
VI.Contribution
The sole author, poses the research question, demonstrates and proves the question.
VII.Author
Name: Teng Liao (1509135693@139.com), Sole author
Setting: Tianzheng International Institute of Mathematics and Physics, Xiamen, China
Work unit address: 237 Airport Road, Weili Community, Huli District, Xiamen City
Zip Code: 361022
References
[1] 《Problems related to flip graph Equation 》；
[2] Riemann : 《On the Number of Prime Numbers Less than a Given Value 》；
[3] John Derbyshire(America): 《PRIME OBSESSION 》P218,BERHARD RIEMANN
AND THE GREATEST UNSOlVED PROBLEM IN MATHMATICS,Translated by Chen
Weifeng, Shanghai Science and Technology Education Press,
China,https://www.doc88.com/p-54887013707687.html;
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