GSTF Journal of Mathematics, Statistics and Operations Research (JMSOR) Vol.2 No.2, October 2014
Proof of Fermat’s Last Theorem for n=3 Using
Tschirnhaus Transformation
B.B.Upeksha P. Perera, and R.A.D. Piyadasa
Abstract— This paper gives a proof on Fermat’s Last Theorem
(FLT) for n=3 by firstly reducing the Fermat’s equation to a
cubic equation of one variable and then using Tschirnhaus
transformation to reduce it to a depressed cubic. By showing that
this last equation has non-rational roots it was concluded that the
Fermat’s equation cannot have integer solutions.
KeywordsFermat’s
transformation; Integer roots
I.
Last
Theorem;
where (x,y) denotes the greatest common divisor of x and y
hence (x,y)=1 meaning that x and y a co-prime.
Following a similar procedure as in [3], consider (2) in the
form
z  y z  y 2  3zy   x 3

Tschirnhaus
and (z,y)=(3,z)=1. Hence we have (z-y) and z  y 2  3 zy 


are co-prime. Hence (z-y) is a cube, say z-y=u3, where u is a
factor of x.
BACKGROUND/JUSTIFICATION
Ever since Pierre de Fermat (1637) left an unfinished
conjecture that the equation
xn  yn  zn

Considering (2) again, in the form
z  x z  x 2  3zx   y 3
(1)


(which is so called Fermat’s Last Theorem or FLT) cannot
have integer solutions for exponent n>2, there has been many
attempts to prove that the statement is true. But until Andrew
Wiles gave a 100 page long proof in 1995 which took him
seven years, it was intractable. In fact, once it was in the
Guinness Book of World Records for most difficult
mathematical problems.
and (z,x)=(3,z)=1. Hence we have (z-x) and z  x 2  3 zx 


are co-prime. Hence (z-x) also is a cube, say z-x =h3, where h
is a factor of y and (u,h)=1.
Recently, there were many shorter proofs for the theorem
(see [4] and [5]) and for the case n=3 (e.g. [1], [2], [3]), in
particular. For example, [1] uses Mean Value Theorem and [3]
uses Method of Infinite Descent.
Thus, x  z  h 3 , and y  z  u 3 . Substituting for x and y in
(2) we get
II.
Thus, we have z  x  h 3 , and z  y  u3 where (u,h)=1.
z 3  (z  u 3 )3  (z  h3 )3
or
OBJECTIVES
z 3  3z 2 (u 3  h 3 )  3z (u 6  h 6 )  (u 9  h 9 )  0
This paper attempts to prove the FLT for the case n=3 using
a different approach which is much simpler and direct. Only
the upper school mathematics is used in solving the problem
and a simple transformation to reduce a cubic equation to a
depressed cubic (a cubic with no quadratic term).
III.
This is a cubic equation in one variable z. Now if we can
show that (3) has no integer roots then the proof follows.
A. Tschirnhaus transformation
Tschirnhaus (1683) showed that a polynomial of degree
n>2 can be reduced to a form in which the xn-1 term have 0
coefficient.
Let Pn(x)=0 be a polynomial equation of order n:
METHOD
This paper gives a proof on Fermat’s Last Theorem (FLT)
for n=3 by firstly reducing the Fermat’s equation to a cubic
equation of one variable. Then using Tschirnhaus
transformation it is reduced to a depressed cubic. By showing
that this last equation has non-rational roots it was concluded
that the Fermat’s equation cannot have integer solutions.
Pn ( x)  a n x n  a n 1 x n 1  ...  a1 x  a 0  0
a n 1
converts Pn into a
na n
n
n 1
 ...  b1 y  b0  0
depressed polynomial: bn y  bn 1 y
where bn 1  0 .
Then the substitution: y  x 
Consider the Fermat’s equation for n=3:
x 3  y 3  z 3 , ( x, y)  1
(3)
(2)
DOI: 10.5176/2251-3388_2.2.45
© 2014 GSTF
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GSTF Journal of Mathematics, Statistics and Operations Research (JMSOR) Vol.2 No.2, October 2014
The cubic equation is a special case of such a
3
2
transformation: ax  bx  cx  d  0 . Dividing equation
t 3  pt  q  0
2b 3  9abc  27 a 2 d
3ac  b 2
q

,
.
27 a 3
3a 2
REFERENCES
[1]
The left hand side of the above equation is a monic
trinomial called a depressed cubic.
[2]
So using Tschirnhaus transformation [6] we can remove the
z term thus reducing (3) to a depressed cubic (or monic
trinomial).
2
Let z  t  u 3  h 3 . This transforms (3) into
[3]
t  6u h t  3u h (u  h )  0
3
3 3
3 3
3
3
(4)
This equation is in the form
t 3  6vwt  v 3  w3  0

(5)

[4]
where v 3  w3  3u 3 h 3 u 3  h 3 and vw  2u 3 h 3 . By using
the method of Tartagalia and Cardano for finding roots of a
cubic equation roots of (5) can be written as
v  w, v  w 2 , v 2  w
[5]
where  is the cube root of unity.
Now, v 3 , w 3 are the roots of the equation
X 2  3u 3 h 3 (u 3  h 3 ) X  8u 9 h 9  0
Roots of (6) are


 3 u 3  h 3  9h 6  14u 3 h 3  9u 6
X  u 3h3 

2

IV.
[6]
(6)




OUTCOME AND APPLICATION
This paper gives a simple and shorter alternative proof for
Fermat’s theorem for n=3 using simple Mathematics. Further
work requires to extend this method to the general exponent n,
and thereby this proof suggests that there is a possibility of
existence of a simple proof to the Fermat’s theorem in general.
b
we get the equation
xt
3a
by a and substituting
where p 
VI.
(7)
[7]
Piyadasa, R.A.D, “Mean value theorem and Fermat's Last Theorem for
n=3,” in the Proceedings of the Annual Research Symposium 2007Faculty of Graduate Studies, University of Kelaniya
Available: http://www.kln.ac.lk/uokr/ARS2007/4.6.pdf[Accessed Nov.
15, 2013].
Piyadasa, R.A.D, “Simple and analytical proof of Fermat's last theorem
for n=3,” in the Proceedings of the Annual Research Symposium 2008Faculty of Graduate Studies, University of Kelaniya
Available: http://www.kln.ac.lk/uokr/ARS2008/4.20.pdf[Accessed Nov.
15, 2013].
Piyadasa, R.A.D, “Method of Infinite Descent and proof of Fermat's last
theorem for n=3,” Canadian Journal on Computing in Mathematics,
Natural Sciences, Engineering & Medicine, 1(6), September 2010.
[Online].
Available:
http://www.ampublisher.com/September%202010/CMNSEM-1009012.pdf [Accessed Sep. 19, 2013].
Piyadasa, R.A.D, “A simple and short analytical proof of Fermat’s last
theorem,” Canadian Journal on Computing in Mathematics, Natural
Sciences, Engineering & Medicine, 2(3), March 2011. [Online].
Available: http://www.ampublisher.com/Mar%202011/CMNSEM-1103015.pdf [Accessed Sep. 19, 2013].
Piyadasa, R.A.D, Shadini, A.M.D.M., Jayasinghe, W.J.M.L.P., “Simple
analytical proofs of three Fermat’s theorems,” Canadian Journal on
Computing in Mathematics, Natural Sciences, Engineering & Medicine,
2(3), March 2011. [Online].
Available: http://www.ampublisher.com/Mar%202011/CMNSEM-1103014.pdf [Accessed Sep. 19, 2013].
Tschirnhaus, E.W., “A method for removing all intermediate terms from
a given equation” ACM SIGSAM Bulletin, 37(1), March 2003
(Translated by R. F. Green) [Online].
Available:http://www.sigsam.org/bulletin/articles/143/tschirnhaus.pdf[A
ccessed Sep. 19, 2013].
Weisstein, Eric W. “Vieta's Substitution.” From MathWorld - A
Wolfram Web Resource. [Online].
Available:http://mathworld.wolfram.com/VietasSubstitution.html
[Accessed Sep. 19, 2013].
RESULTS
By observing that the expression inside the square root can
be written as
AUTHORS’ PROFILE
2
7 
32 6

9h 6  14u 3 h 3  9u 6   3h 3  u 3  
u 0
3 
9

B.B.U.P. Perera is a Lecturer in Mathematics at University of Kelaniya,
Sri Lanka. She did her B.Sc. (Special) in Mathematics in 2006 from
University of Ruhuna, Sri Lanka with first class honors. Her research
interests are Neural Networks, Number Theory and Homogeneous
Programming.
Dr. R.A.D. Piyadasa is a Senior Lecturer (Grade I) in Mathematics at
University of Kelaniya, Sri Lanka. He did his B.Sc. (Special) in
Mathematics from University of Kelaniya, Sri Lanka and obtained M.Sc.
and Ph.D. from Kyushu University, Japan. He won Presidential award
for scientific research in 2001 and Vice-chancellor's award for bringing
honor to the University in 2006. His research interests are Quantum
mechanics, Quantum Field theory and Number Theory.
which is never zero or a perfect square for all non-zero u and h
we can see that the two roots v 3 , w 3 of (6) are irrational.
V.
CONCLUSIONS
Therefore the only real root of (5), v  w is also irrational
so that t and in turn z also is irrational meaning that the FLT for
n=3 cannot possibly have integer solutions.
© 2014 GSTF
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