Fermat’s Last Theorem has been proved

Jiang, Chun-xuan

P. O. Box 3924, Beijing

The People’s Republic of China

( This paper became the most important reason of the four reasons Jiang Chun-xuan received the Telesio-Galilei 2009 gold medal award)

（本文为蒋春暄荣获2009年Telesio-Galilei金奖四个理由中最重要的理由）

http://www.telesio-galilei.com/awards2009.html

Note: The text of Jiang’s above mentioned Preprint of “Fermat’s Last Theorem has been proved” in English, mailed to over 600 mathematicians worldwide in January 1992, is exactly the same with Jiang’s paper Fermat’s Last Theorem has been proved, published by Potential science (Chinese) No. 2, 1992, 17-20.

注： 中国《潜科学》杂志1992年第2期17-20页上发表的蒋春暄中文《费马大定理已被证明》与蒋春暄1992年1月寄给世界600多位数学家的英文“Fermat’s Last Theorem has been proved”英文预印本文本内容完全一致。

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In this paper we prove that it is sufficient to prove for Fermat’s last theorem using the complex hyperbolic functions in the hypercomplex variable theory. More than 200 years ago Euler gave a proof of . Fermat’s last theorem has been proved.

First we discuss the hyperbolic functions

(1)

From (1) we get its inverse transformation

(2)

From (2) we get

(3)

If =rational, then (3) has infinitely many rational solutions.

Using above method we prove Fermat’s last theorem. From Refs 1 and 2 we introduce the complex hyperbolic functions of order p:

where , (4)

. (5)

(6)

From (4) we get its inverse transformation^[3,4]

(7)

, (8)

, (9)

Assume , , . From (7)-(9) we get

, (10)

, (11)

, (12)

From (11) and (12) we get

, (13)

, (14)

From (5) we get

(15)

From (10), (13), (14) and (15) we may get Fermat’s equations

, (16)

(17)

………

(18)

Theorem. (18) has no rational solutions for , except .

Proof 1. Since are the funcitons of , we study in (5). Put , where odd prime. From (5), (10), (13), (14) and (15) we get

, (19)

, (20)

, (21)

(22)

(23)

(19), (20) and (22) are the Fermat’s equations. Since Euler gave a proof of (20), (19). Therefore and (22) have no rational solutions for any odd prime .

From (22) and (23) we get

,

. (24)

If rational, then irrational and vice versa. Therefore (19), (20) and (22) have no rational solutions for any odd prime .

From (22) and (24) we get

(25)

where

,

(26)

If , they are two trivial solutions. If rational, then irrational and vice versa. Therefore (25) has no rational solutions for any odd prime .

Proof 2. Put , where =odd prime. From (5), (10), (13), (14) and (15) we get

, (27)

, (28)

(29)

(30)

(31)

(27)-(31) are the Fermat’s equations. Since Euler gave a proof of (28), therefore (27), (29), (30) and (31) have no rational solutions for any odd prime .

Example. From (5) we get

(32)

where

From (10), (13), (14), (15) and (32) we get

, (33)

, (34)

. (35)

(33)-(35) are the Fermat’s equations. Since Euler gave a proof of (34), therefore (33) and (35) have no rational solutions.

Remark. In order to prove Fermat’s last theorem, it is sufficient to prove and . The former was due to Fermat and the latter was due to Euler. Fermat’s last theorem has been proved. On Oct. 25, 1991 without using any number theory we have proved Fermat last theorem.

References

1. Jiang, Chun-xuan. Hypercomplex variable theory. Proeprints, 1989.

2. Jiang, Chun-xuan. Hypercomplex variable theory and FLT (I). Preprints, 1991.

3. Jiang, Chun-xuan. Nonlinear dynamics in Santilli’s eletonic model of hadronic structure. Hadronic J. 11(1988) 125-142.

4. Jiang, Chun-xuan. The dynamics for the prime principle. Acta Math. Sci. 9(1989) 93-107. MR 90k: 58101.

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蒋春暄按：

Remarks by Jiang Chun-xuan:

中国著名数论专家乐茂华教授看到本文1992年1月21日来信“至于您这次寄来的文章，不知已在何种刊物发表？望来信告知。”他是第一位看到本文的专家，对本文肯定，以后他多次来信支持蒋春暄的工作，他是广东湛江师范学院教授，1992年本文已在全世界数学家散发600份, 中科院数论家都看到本文。

Prof. Le Mao-hua, the outstanding Chinese number theory specialist, after reviewing this paper he wrote a letter on Jan. 21, 1992 to Jiang stating: “Regarding the paper you mailed this time, it is unknown on which journal it is published? Please write and inform me.” He is the first specialist to read this paper, and affirmed it. He later wrote many letters encouraging Jiang’s work. He is professor at the Guangdong Zhanjiang Normal University. Over 600 copies of this paper were mailed to mathematicians all over the world, the number theory mathematicians of the China Academy of Science have all seen this paper.

十七年后本文成为蒋春暄荣获2009年度Telesio-Galilei金奖四个理由中最重要的理由。重读这篇论文，蒋春暄认为他最大贡献是证明费马大定理（1992），这是天才的证明，直接的证明,伟大的证证明，绝后的证明，费马的证明，最高级的证明, 四页的证明。1995年怀尔斯(Wiles proof of Fermat Last Theorem)200多页的费马大定理证明是可怜的，间接的证明,没有意义的，张冠李戴的，无法理解的，骗人的，最低级的,200页的证明。

Seventeen years later it becomes the most important reason of the four reasons Jiang Chun-xuan received the Telesio-Galilei 2009 gold medal award. Reviewing this paper again, Jiang Chun-xuan considers: his greatest contribution is the proof of Fermat’s Last Theorem (1992), which is a genius proof, a direct proof, a great proof, a proof never to be seen again, Fermat’s proof, most senior level proof, an only four pages proof. Wile’s proof of Fermat Last Theorem is a pitiful indirect proof, it has no significance, it confuses one thing with another, unable to understand, deceptious, low class, over 200 pages proof.

不同意吗？那么请首先否定我1992年公开发表的上述证明！

Do not agree? Then please first disprove my above proof published in 1992!

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Remarks by Chen I-wan, scholar of Innovative Science Sociology:

科技创新社会学学者陈一文按语：

Jiang’s proof of FLT was published in China in March 1992 (Jiang Chun-xuan, Fermat’s Last Theorem has been proved, Potential Science, 2, 17-20 (1992)); and later published in English in the USA in 1994 (Jiang Chun-xuan, Algebras, Groups and Geometries, 11, 371-377 (1994)).

蒋春暄对于费马大定理的证明1992年3月在中国发表〔蒋春暄，费马大定理已被证明，潜科学，2，17-20〔1992〕；而后于1994年在美国以英文出版〔蒋春暄，代数·群·几何，11, 371-377〔1994〕〕。

In early 1992 and again in 1993, Jiang mailed over 600 copies of preprints of Jiang’s proof of FLT to numerous mathematicians in China and the world, including the Princeton University where Wiles worked. Jiang accomplished all of this was far before Wiles made his final announcement that he has eventually proved FLT in 1995.

蒋春暄于1992年初并再次于1993年将600多份蒋春暄对于费马大定理的证明的预印本邮寄发给中国与世界无数数学家，包括怀尔斯工作的普林斯敦大学。蒋春暄实现所有这些远在怀尔斯1995宣布自己最终证明费马大定理之前。

The text of Jiang’s above mentioned Preprint in English, mailed to over 600 mathematicians worldwide in January 1992, is exactly the same with Jiang’s paper Fermat’s Last Theorem has been proved, published by Potential science (Chinese) No. 2, 1992, 17-20

中国《潜科学》杂志1992年第2期17-20页上发表的蒋春暄中文《费马大定理已被证明》与蒋春暄1992年1月寄给世界600多位数学家的英文“Fermat’s Last Theorem has been proved”英文预印本文本内容完全一致。

Learning that the Prof. Yang Zhen-ning, on behalf of the Shao Yi-fu Science Prize Foundation, and supported by the China Mathematics Society, was going to present the Shao Yi-fu 2005 Science Gold Award to Andrew Wiles for being “The first to prove Fermat’s Last Theorem”, advisor Chen I-wan wrote An Open Letter dated to Prof. Yang Zhen-ning, The Shao Yi-fu Science Prize Foundation, and to The China Mathematics Society on Aug. 16, 2005, and posted the remarks below:

获悉杨振宁教授在中国数学会的支持下代表邵逸夫科学奖基金会将把2005年度邵逸夫科学金奖因“第一个证明费马大定理”授予安德鲁斯·怀尔斯，陈一文顾问于2005年8月6日发表了《致杨振宁教授、邵逸夫科学奖基金会与中国数学学会的公开信》，同时发表按语如下：

Therefore, if Wiles, and/or other mathematicians abroad or in China supporting Wiles, claim that Wiles was the first in the world to prove FLT, then, Wiles, or they, must first disprove Jiang Chun-xuan’s above mentioned prove of FLT published in early 1992.

因而，如果怀尔斯，和／或支持怀尔斯的任何其它国外或中国的数学家，声称怀尔斯为世界上第一个证明费马大定理的人，那么，怀尔斯，或者他们，必须首先否定1992年初已经发表的蒋春暄对于费马大定理的证明。

Failing to disprove Jiang Chun-xuan’s prove of FLT published in early 1992 and mailed to Wiles and over six hundred mathematicians and institutions in China and worldwide, then Wiles, and/or all other mathematicians abroad or in China supporting Wiles, are not qualified to claim or agree that Wiles was the first in the world to prove FLT and are on this issue not only cheating the world!

如若未能否定蒋春暄1992年初就已经发表并寄给世界各地600多位数学家与机构的蒋春暄对于费马大定理的证明，则怀尔斯，和／或世界各地与中国支持怀尔斯的所有数学家，没有资格声称或赞同怀尔斯为世界上第一个证明费马大定理的人，且在该问题上不仅欺骗全世界!

If Wiles, and/or all other mathematicians abroad or in China supporting Wiles, do not agree with the logic of my above statement, then please present your proof disproving the logic of my above statement. Otherwise it only proves that you are also cheating yourself!

如若怀尔斯，和／或世界各地与中国支持怀尔斯的所有数学家，不同意我的上述声明的逻辑性，请你们拿出否定我的上述声明逻辑性的证明！否则只能证明你们也在自我欺骗！]

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