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 蒋春暄 （jiangchunxuan@sohu.com） 2008.03.23 19:52:59
Reapperance of Fermat’s Marvelous Proofs
for Fermat’s Last Theorem
JIANG Chun-xuan
( Beijing System Design
Institute of Electro-Mechanic Engineering, P. O. Box 3924, Beijing 100854, P.
R. China)
jiangchunxuan@sohu.com
Abstract: Using the
complex hyperbolic functions and complex trigonometric functions we reappear
the Fermat’s marvelous proofs for Fermat’s last theorem (FLT) [1-7]. We
present three proofs: (1) Jiang’s marvelous proofs, (2) Fermat’s marvelous
proofs and (3) Frey-Ribet-Wiles proofs. Ribenbiom [11] points out that there
are some mathematicians who are not satisfied with the method of proof using
elliptic curves and modular form, perhaps wrongly? or rightly?
Key words: Fermat last theorem; Marvelous proofs.
MR(1991) Subject
Classification: 11D41/CLC number: O156. 2
1. Introduction
In 1637, the reading of Diophantus’ Arithmetica,
in particular, the part on the Pythagorean equation, inspired Fermat to write
in his copy of Diophantus’ monograph:
It is impossible for a cube to be written as a sum of two
cubes or a fourth power to be written as the sum of two fourth powers or, in
general, for any number which is a power greater than the second to be written
as a sum of two like powers. I have a truly marvelous demonstration of this
proposition which this margin is too narrow to contain.
Fermat never published a proof and, by the
unsuccessful quest for a solution of Fermat’s last theorem, mathematicians started
to believe the that Fermat actually had no proof. However, no counterexample
was found..
In this paper using the complex hyperbolic
functions and complex trigonometric functions we reappear Fermat’s marvelous
proofs for FLT. Fermat proved that there are no integral solutons for the FLT
exponent  ,
Euler proved FLT exponent  .
2. Jiang’s
Marvelous Proofs
Theorem 1.
It is sufficient to prove that the FLT exponents  are odd primes. But this proof
has great difficulty. We consider that FLT exponents are the composite
numbers. Let  ,
where  is
an odd prime. Using the complex hyperbolic functions we have the Fermat’s
equations [2, 5, 7]
(1) 
(2)  .
Fermat proved (1), therefore (2) has no
rational soultions for any odd prime .
Note. Let  . Every factor of
the FLT exponent has a Fermat’s equation [1-7].
Theorem 2.
Let  ,
where  is
an odd prime. Using the complex hyperbolic functions we have the Fermat’s
equations [1, 3, 4, 5, 6, 7]
(3) 
(4)  ,
(5)  .
Euler proved (3) and (4), therefore (5) has
no rational solutions for any odd prime .
Note. Let  . Every factor of
the FLT exponent has a Fermat’s equation [1-7].
3. Fermat’s
Marvelous Proofs
Theorem 3.
Fermat’s equation
(6) 
has no integral soultions  with  , if  . We assume that
if  and
 are
integral numbers, then  is irrational numbers.
Proof 1. Let
, where is an odd
prime. From (6) we have the Fermat’s equations
(7) 
(8) 
(9) 
Since Fermat proved the FLT exponent , we prove that
(7) and (8) hae no integral solutions, that is and  are irrational
numbers. We prove that (9) has no integral solutions for any odd prime , that is  is
irrational numbers.
We rewrite (8) and (9) as
(10) 
(11) 
where  , .
Fermat proved (7) and (10), therefore (11)
has no integral solutions for any odd prime .
Note. Let  . Every factor of
the FLT exponent has a Fermat’s equation [1-7].
Proof 2. Let
, where is an odd
prime. From (6) we have the Fermat’s equations
(12) 
(13) 
(14) 
Since Euler proved the FLT exponent , therefore (12)
and (13) have no integral solutions, that is and are irrational
numbers. We prove (14) has no integral solutions, that is  is irrational number.
We rewrite (13) and (14) as
(15) 
(16) 
where  , .
Euler proved (12) and (15), therefore (16)
has no integral solutions for any odd prime , that is C and D are
irrational numbers.
Note. Let  . Every factor of
the FLT exponent has a Fermat’s equation [1-7].
4. Frey-Ribet-Wiles Proofs
(Ⅰ) From elliotic curve to Fermat’s equation
Using elliptic curve we prove FLT. We
discuss Fermat’s equation
(17) 
integral solutions. Frey [8] write (17)
Fermat’s equation as elliptic curve
(18)  .
He conjectures that (18) elliptic curve
would imply (17) Fermat’s equation. Ribet [9] prove that (18) elliptic curve
should imply (17) Fermat’s equation. Wiles [10] prove that (18) elliptic curve over
Q is modular. But he does not discuss and prove (17) Fermat’s equation.
Note.
Fermat’s equation is th power, but elliptic curve is 3th
power. This proof is incredible to number theorists. Ribenboim [11] points out
that there are some mathematicians who are not satisfied with the method of
proof using elliptic curves and modular form, perhaps wrongly ? or rightly ?
(Ⅱ) From Fermat’s equation to elliptic curve
Using Fermat’s equation we prove elliptic
curve. We discuss elliptic curve
(19) 
integral solutions. Fabs write (19)
elliptic curve as Fermat’s equation
(20)  .
He conjectures that (20) Fermat’s equation
would imply (19) elliptic curve. Rabs prove that (20) Fermat’s equation should
imply (19) elliptic curve. Jiang [1-7] prove that (20) Fermat’s equation no
integral solutions. But he does not discuss and prove (19) elliptic curve.
Note. (I)
and (II) cases are the same. Both proofs are incredible to mathematicians.
References
[1] 蒋春暄，费马大定理已被证明，潜科学，2，17-20（1992）
[2] 蒋春暄，三百多年前费马大定理已被证明，潜科学，6，18-20（1992）
[3] Chun-Xuan, Jiang, On the factorization theorem of circulant
determinant, Algeras, Groups and Geometries, 11, 371-377 (1994).
[4] Chun-Xuan, Jiang, Fermat’s last theorem was proved in 1991,
Preprints (1993), In: Foundamental open problems in science at the end of the
millennium, T. Gill, K. Liu and E. Trell, eds, 555-558 (1999).
[5] Chun-Xuan, Jiang, On the Fermat-Santilli isotheorem, Algebras,
Groups and Geometries, 15, 319-349 (1998).
[6] Chun-Xuan, Jiang, Complex hyperbolic functions and Fermat’s last
theorem, Hadronic journal supplement, 15, 341-348 (2000).
[7] Chun-Xuan, Jiang, Foundations of Santilli’s isonumber theory
with applications to new cryptograms, Fermat’s theorem and Goldbach’s
conjecture, pp. 225-263, Inter. Acad. Press, 2002, MR2004c: 11001, http: // www.i-b-r.org/docs/Jiang/pdf.
[8] G. Frey, Links between stable elliptic curve and certain Diophantine
equations, Annales Universitatis Sarviensis 1, 1-40 (1986).
[9] K. A. Ribet, On modular representations of Gal ( ) arising from
modular forms, Invent. Math. 100, 431-476 (1990).
[10] A. Wiles, Modular elliptic curves and Fermat’s last theorem,
Ann. Math. 141, 443-551 (1995).
[11] Paulo Ribenboim, Fermat’s last theorem for amateurs,
Spring-Verlag, New York, 1999, pp.366.
重现费马大定理费马奇妙证明
蒋春暄
（北京机电工程总体设计部，北京，100854）
摘要：采用复双曲函数和复三角函数我们重现费马大定理费马奇妙证明[1-7]。我们讨论三种证明：（1）蒋奇妙证明，（2）Fermat奇妙证明，（3）Frey-Ribet-Wiles证明。Ribenbiom[11]指出有些数学家不满意采用椭园曲线和模型证明方法，可能是错？也可能是对？
关键词：费马大定理；奇妙证明
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