Fermat’s Last Theorem is that for each&nbsp; k &gt;=3&nbsp; (k is an integer), the equation &nbsp;involving x, y and z has no positive integer solution. This paper proposed a novel proof for the Fermat’s last theorem by the methods of infinite descent and complex variable analysis

Cai, Ling

Chen, Songlin

Zhang, Qing

English

Scitech Research Journals

Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1712|        SCITECH                                                                        Volume 12, Issue 1 RESEARCH ORGANISATION                   Published online: April 18, 2017|        Journal of Progressive Research in Mathematics   www.scitecresearch.com/journals     A Novel Proof based on the method of infinite descent for Fermat’s Last Theorem  1Cai Ling, 2Chen Songlin*, 3Zhang Qing  1 School of Mathematics & Physics, Anhui University of Technology, Ma’anshan, Anhui 243002, P R China.   E-mail address: clagd@163.com 2* School of Mathematics & Physics, Anhui University of Technology, Ma’anshan, Anhui 243002, P R China.  Corresponding author.   E-mail address: slchen@ahut.edu.cn  3School of Mathematics & Physics, Anhui University of Technology, Ma’anshan, Anhui 243002, P R China.   E-mail address: zhqing@ahut.edu.cn  Abstract     Fermat’s Last Theorem is that for each  k3  (k is an integer), the equation k k kx y z   involving ,x y  and z has no positive integer solution. This paper proposed a novel proof for the Fermat’s last theorem by the methods of infinite descent and complex variable analysis. Keywords: Fermat’s Last Theorem; elementary proof; the method of infinite descent.  1. Introduction P.de Fermat(1601─1665) wrote the famous theorem in the margin of a book in 1637 and said that he had found a very ingenious proof but the margin is too small to write it. There are some other statements included in [1]. For three and half centuries, searching for the proof, especially, a succinct proof of the theorem has puzzled and encouraged numerous mathematicians. After a century, L.Euler (1707─1783) and C.F.Gauss (1777─1855) gave the proof successively for k=3. Recently, A. Wiles [2,3] proved the theorem based on advanced algebra and cyclic groups. The present paper devotes to propose an ingenious and succinct proof for Fermat’s last theorem by complex analysis[4] and the method of infinite descent that had been interested by Fermat.  The remaining part of this paper is organized as follows. In section 2 the principle of infinite descent for the Fermat’s last theorem is set up by complex variable analysis. In section 3 we prove the case of k=3 for Fermat’s last theorem directly by the ideas in section 2. The complete proof of Fermat’s last theorem is integrated in section 4.  2. The principle of infinite descent for the Fermat’s last theorem  Lemma 1.  For the equation wherein x, y, z, k are positive integers                                              k k kx y z                                                                                           (1)  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1713|          If for a integer k 3, there exists a positive integer solution (x,y,z), then the equation (1) with k replaced by k-1 has also a positive integer solution.    Lemma 1 is called the principle of infinite descent for the Fermat’s last theorem. Proof   We set                    21nxznxy                                                                                                                                  (2) wherein  n and n are positive integers less than x.  Substituting (2) into (1), we have                     kjjkjjjkjk xnnCx121 0)()1(                                                                    (3) in which  ])!(![! jkjkC jk  . The k roots of (3) correspond with the k points in complex plane (The roots may be multiple).  Denote the roots as lill eax    or                   .k,,2,1l),sini(cosax llll                                                                                         (4) in which la  and l are real numbers ( 0 )l la ,      .                                                         Substituting (4) into (3) and separating the real and imaginary part, we obtain a pair of equations for each l as follows:                         k.,1,2,l    0jksinannC1sinka0jkcosannC1coska1k1jljklj2j1jkjlklk1jljklj2j1jkjlkl])[()()(])[()()(              (5) It is well known that just these k points in complex plane can make the (5) hold. Note that in the second equation of (5), summation for j is from 1 up to k-1 as the k-th term is zero.  The sine and cosine functions in (5) can be expanded as power series, thus we have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1714|        k1,2...,l]5!j)(k3!j)(k[)an(nC1)(]5!k3!k[aj)(k)an(nC1)(ka]4!j)(k2!j)(k[)an(nC1)(]4!k2!k[a)an(nC1)(a1k1j4l52l3jklj2j1jkj4l52l3kl1k1jjklj2j1jkjklk1j4l42l2jklj2j1jkj4l42l2klk1jjklj2j1jkjkl(6)  Among the k roots of (3), we assume there are m roots ( km  ) of which modules are positive numbers (signed as bl) and polar angles  l  (  is an infinitesimal quantity). Thus, we can change (6) into m pairs of equation corresponding with the m roots as below:  ).(,,2,1!5])()()1([!3])()()1([])()1([!4])()()1([!2])()()1([)()1(11115)1(2114143)1(21121211)1(211114)(214412)(2122121kmmlbnncjkbkbnncjkbkbnnCbbnncjkbkbnnCjkbkbnnCbkjkjljkljjjkjklljkljjjkjkllkjjkljjjkjklkjljkljjjkjklkjljkljjjkjklkjjkljjjkjkl                                                                                                                                                              (7)  Considering the infinitesimal neighborhood ( lil eb ) of the m points ( lil eb ),which is the m roots of (3) in complex plane.  We suppose 0l  , ll    and demand ll    be small enough.  Substituted lil ebfor lil eb,  the (7) are no more exactly equations. The first and second expression of (7) becomes second and third order infinitesimal quantity respectively. Thus we have: k),m.(m,2,1l)(o]b)nn(c)1(b[)(ob)nn(c)1(b3l1k1jlj)1k(lj2j1j1kj1klk1j2ljklj2j1jkjkl                                     (8)  in which )(o 2l and )(o3l  are respectively the second and third order infinitesimal quantity. Among the m roots, we suppose there are s roots ( ms  ) at the real axis. For the neighborhood of the s points, (8)still holds. Corresponding with the neighborhood of the s points, we have:         .  m),s.(s,2,1 l)(ob)nn(c)1(b)(ob)nn(c)1(b2l1k1jj)1k(lj2j1j1kj1klk1j2ljklj2j1jkjkl                                                         (9) Furthermore, limiting the neighborhood of the s points to the real axis, i.e. 0l  , we obtain s pairs of equation from (9): Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1715|          .  m),s.(s,2,1 l0b)nn(c)1(b0b)nn(c)1(b1k1jj)1k(lj2j1j1kj1klk1jjklj2j1jkjkl                                          (10) (10) shows if lb  is a positive integer, only these s roots of (3) at real axis can make (10) hold. So far, we can see if (3) has positive integer number solution, the (10) must hold.   Once more, we consider equation:                        1k1k1k zyx                                                                                                               (11) Analogously, we set                        21nxznxy                                                                                                                      (12) where 1n  and 2n  are some positive integers less than x. Substituting (12) into (11) gives:                        1k1jj)1k(j2j1j1kj1k 0x)nn(c)1(x                                                                          (13) By using of the combination identity                        jkjk kCjkC 1)(   We have from (13)                       11 ( 1)1 1 2111 ( 1)1 21( 1) ( )( 1) ( ) 0kk j j j j k jkjkk j j j j k jkjkx kC n n xkx k j C n n x                                                               (14) Now, suppose ,on the contrary, (11) has no positive integer solution,then for arbitrary positive integer x, n’1 , n’2 , we obtain from (13):                      11 ( 1)1 1 21( 1) ( ) 0kk j j j j k jkjx C n n x                                                                            (15)  Because 1n , 2n , 1n , 2n  are arbitrary positive integers, the second equation of (7) is of the same pattern as the equation (14). If (15) holds, the second expression of (7) will not hold. Thus, (1) has no positive integer solution for index k if it has no positive integer solution for index 1k-  .   3. The proof for the case of k=3 in Fermat’s last theorem Lemma 2.   For the equation wherein x, y, z are positive integers  3 3 3x y z     Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1716|          There exists not a positive integer solution (x, y, z). Proof.  According to the equations (7) obtained above, it follows         3 2 2 2 3 31 2 1 2 1 22 2 21 2 1 23 3 02 0b n n b n n b n nb b n n n n                                                  (16)   Combine two equations in (16) results in                     2 2 2 3 31 2 1 2 1 22 0n n b n n b n n        Hence we obtain with (16b)                  1 24b n n                                                                                                                       (17)  On the other hand, (16b) is a quadratic equation, it has roots                  1 2 1 22b n n n n                                                                                                       (18)  b,n1,n2 being positive, thus we have from (17) and (18)                 2 21 1 2 29 14 9 0n n n n                                                                                                    (19) The =196-324=-128<0 for (19) means that there are no positive n1, n2, thus b, to satisfy (16) . 4. The Proof for the Fermat’s Last Theorem Theorem (Fermat’s last theorem)   For the equation wherein x, y, z, k are positive integers  k k kx y z       k 3. there exists not a positive integer solution (x,y,z).   Proof.  Combining the results of the principle of infinite descent for the Fermat’s last theorem in §2 and the case of k=3 for Fermat’s last theorem in §3 obtained above, we obtain that (1) has no positive integer solution for arbitrary positive integer k 3, i.e. Fermat’s last theorem holds.     References [1] H. Edwards (1977).Fermat's Last Theorem: A Genetic Introduction to Algebraic Number theory. Springer-Verlag, New York. [2] A.Wiles(1995). Modular elliptic curves and Fermat's Last Theorem. Ann. Math., 141, 443-551. [3] A.Wiles & R.Taylor(1995). Ring theoretic properties of certain Heche algebras. Ann. Math., 141,553-573. [4] M. Beck, G. Marchesi, D. Pixton, L. Sabalka 2002.  A First Course of Complex Analysis. Orthogonal Publishing. [5] J.E. Joseph(2016).Proofs of Fermat's Last Theorem and Beal's Conjecture. JPRM, 10, 1446-1447. Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218     Volume 12, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                1717|        Acknowledgments The authors acknowledge the fund for the research supported by the Natural Science Foundation of Education Department of Anhui province (No. KJ2016A084) The authors express their acknowledgement to Prof. Guangjiong Ni(Fudan University), Prof. Renjie Mao(Shanghai Jiaotong University),  Prof. Qu Li(Shanghai Jiaotong University),  Prof. Xiaoshuang Chen(ShanghaiInstitute of Technical physics, Chinese Academy of Science), and the anonymous reviewers for their good suggestions about the paper. 

A Novel Proof based on the method of infinite descent for Fermat’s Last Theorem

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A Novel Proof based on the method of infinite descent for Fermat’s Last Theorem

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