Simple Algebraic Proofs of Fermat s Last Theorem. Samuel Bonaya Buya*

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1 Avaiabe onine at eagia Research Library Advances in Appied Science Research, 017, 8(3:60-6 ISSN : CODEN (USA: AASRFC Simpe Agebraic roofs of Fermat s Last Theorem Samue Bonaya Buya* Ngao Girs Secondary Schoo, Kenya ABSTRACT In this research simpe proofs of Fermat s ast theorem of ast theorem are proposed. The proofs presented do not require any Gaois representation or concepts of eiptic curves. In the first proof a most genera agebraic coordinate representation of ythagorean integer tripes is proposed. The tripes wi be powered to some genera degree n to enabe derivation of the proof. In the second proof method a genera agebraic formua containing a the ythagorean tripes is proposed. The formua is then used to prove Fermat s ast theorem. The mathematics in this proposed agebraic form is trivia and within the scope of seventeenth century mathematics. Fermat caimed that he got a tremendous proof of his theorem. The objective of this paper is to show that there are simpe mathematica proofs of Fermat s ast theorem within the reach of the seventeenth century mathematics. Keywords: roof of Fermat's ast theorem, ythagorean tripes HISTORY AND OVERVIEW The integer soution of x + y = z is we known. Exampes of we-known tripets that satisfy the identity above are (3,, 5; (5, 1, 13. The Babyonians were aware of many different kinds of ythagorean integer tripets. In 1637 ierre de Fermat caimed that the Diophantine equation x + y = z has no soution for any N greater than. Fermat excaimed caimed that he got a marveous proof of his proposition. The proof of this statement has euded mathematicians for centuries. The first compete proof of Fermat s ast theorem for case N=3 was given Kar Friedrich Gauss. eter Dirichet and Andrien Legendre roved Fermat s ast theorem for the case N=5 in 185. Gabrie Laḿe proved Fermat s ast theorem for the case N=7 around Between 187 and 1853 Ernest Kummer pubished some masterfu piece of work in which he attempted the extent to which the function: { ao a1 a ai Z} Z{ ζ] = + ζ ζ : ζ = e πi (ι rime is a unique factorization domain (UFD The basic idea in the proof of Fermat s ast theory is to show is to factor and to show there is no soution of with Λ xyz [1]. x + y = z as ( 1 ( ζ...( ζ x+ y x+ y x+ y = z In 183 Sophie Germaine estabished a proof of conditions under which Fermat ast theorem has no soution. Arthur Wieferich aso proved conditions under which Fermat s ast theorem has no soution. eagia Research Library 60

2 Buya AdvApp Sci Res., 017, 8(3:60-6 There are many who contributed towards the modern proof of Fermat s ast theorem incuding Srinivasa Ramanujan, Andre Wei and John Tate among others. Study of Gaois representation s foowed from the work Andre Wei and John Tate invoving the study of eiptic curves of the form y = g( x On th October 199, Wies produced a manuscript which was vetted and pubished in May 1995 in which the moduarity theorem was estabished as the ast step in proving Fermat s ast theorem. In this paper attempts wi be made to present two simpe proofs of Fermat s ast theorem. In the first proof method an attempt wi be made to present a most genera form of the ythagorean tripets by an attempt wi be made to generaize them to the form a + b = c stricty for the cases N> In the second method I wi seek to present an nth degree poynomia equation by which I wi attempt to generaize ythagorean tripes for the case N> [-7]. METHOD 1 Many formuae for generating ythagorean tripes have been deveoped. Eucid s formua is known to be very successfu in generating many primitive tripes. Eucid s formua fais to produce a tripes. A ythagorean tripe representation wi be presented that takes care of a tripes. A very imited representation of the ythagorean tripes can be represented in the three ordered pair of co-ordinates beow. (N + 1 (M + 1 (N (M ( k(m 1(N 1, k, k Where k, N are integers With k=1 and N=5 000 for exampe we can construct the tripet: (10 001, , that satisfies the above identity. A more genera construction of ythagorean integer tripets can take the form: (N + 1 (M + 1 (N (M + 1 ( k(m + 1(N + 1, k, k 0. Where k, M and N are integers The tripet (8, 55, 73 can be constructed using k=1 M=5 and N=. The most genera construction of ythagorean integer tripets takes the form: ( + ( + ( + 1 ( + 1 ( ( + 1 M N M N ( k M 1 N 1, k, k 0.3 For positive integers in the tripet ( M + 1 > ( N + 1, M, N,, and k are integers. If we seect k=1, M=3, N= and ==5 we obtain the tripet the ythagorean tripet ( , , Notice the sum of the tripes 0.3 is given by ( 1 p ( 1 ( 1 s k N M N = For positive integers, k, M, N we can identify the foowing properties (from 0. of ythagorean tripes: 1. ositive Integers of any given ythagorean tripe add up to an even number.. Odd positive integers in any ythagorean tripe appear in a pair. 3. For any ythagorean positive integer tripe with an odd coordinate, N + 1, there exists at east some corresponding odd coordinate N + N + 1 and even coordinate N + N + 1 to make up the compete tripe.. The sum the coordinates of a ythagorean tripe with odd coordinates, (N+1, N + N + 1is (N+1 (N+ eagia Research Library 61

3 Buya AdvApp Sci Res., 017, 8(3: The coordinates N + N, N + N and n n n a + b = c of a ythagorean tripe are co-prime. The generaization of the concept of ythagorean tripes is the search of positive integers a, b and c such that n n n a + b = c for some n stricty greater than. The tripets 0.3 are the most appropriate for such a generaization since it contains a integer ythagorean tripes and is in power form. If we take (M+1 = a 0.5 (N+1 = b 0.5 The generaization of ythagorean tripes 0.3 woud take the ythagorean form: a b a + b a b + = 0.6 The expansion rearrangement and simpification of the generaization 0.3 above woud ead to the resut beow: a + b = a + b 0.7 If we take = Then the equation 0.7 takes the identity form: a + b = a + b 0.8 The basic idea in the proof of Fermat s ast theory is to show is to factor x + y = z for > We fai to achieve such a generaization with the most genera ythagorean tripes reationship. We end up with a staemate condition. In equation 0. if we take: x = k M + 1 N ( M 1 ( N 1 y k + + = ( M 1 ( N 1 z k = k,, M, N,, are integers. Then the tripets 0.3 take the ythagorean form x + y = z 1. = x k M N ( M + 1 ( N y k k M N (( 1 ( 1 = = + + ( M ( N z k k M N (( 1 ( 1 = = eagia Research Library 6

4 Buya AdvApp Sci Res., 017, 8(3:60-6 x, y and z are aways integers when = this is because ( 1 k M + ( N + 1 and are even numbers and ( + + are even numbers and 1 1 = k M 1 N 1 Therefore, x k ( M ( N = , 1 y k ( M 1 ( N 1 1 ( = are whoe numbers. z k M N Consider equations ; ( If we take k = ((M (N + 1 Then = + + and x= M N + 1 M + 1 N (( M+1 - ( N y= (( M+1 + ( N+1 (( M+1 -( N+1 = x M+1 N ( 1 ( 1 1 z = M + + N Again equations wi aways resut in integer vaues of x, y and z when =. Dividing 1.8 to 1.7 we get the foowing resuts: (( ( 1 M+1 - N+1 1 M+1 N+1 y x = = M+1 N N+1 M+1 > the quotient 1.9 is a surd (exceptiona case where M=N=0 in which case the quotient is zero. This is in which y=0. This means for greater than either x or y or both a radica number. This means that when >, x, y and z cannot simutaneousy yied integer vaues, except in the specia case in which either x or y is zero. Thus Fermat s ast theorem is proved. There is need for another confirmatory proof. METHOD Consider the agebraic equation: n n n n n (n n n n ( x+ m + x + 1 Cx ( x+ m + Cx ( x+ m n1 Cx ( x+ m = z 0.1 Its factorized form is given by: ( n x m x z n + + = 0. (Note here that the equation 0. further simpifies to ( x+ m + x = z a case true for n=1 Consider the imit case of 0.1 n n n x+ m + x = z 0.3a If we take = n 0.3a takes the form beow eagia Research Library 63

5 Buya AdvApp Sci Res., 017, 8(3:60-6 x+ m + x = z 0.3b In such a case Cx ( x + m + Cx ( x + m Cx ( x + m = 0 0. n n n 1 n 1 Equation 0.1 becomes equa to zero ony when x+m=0 When x+m, then the Diophantine equation 0.3 takes the form ( 0 + x = z 0.5 Thus for the genera case >, x+m=0 and equation 0.3 takes the form 0.5 This therefore authenticates Fermat s ast theorem. SUMMARY, CONCLUSION AND RECOMMENDATIONS Simpe methods for proving Fermat ast theorem do exist. It is very possibe for Fermat to come up with one of such proofs given that mathematics had deveoped enough in his time to come with such a proof. The genera forms of deriving the ythagorean tripes can be used to computer generate infinite number of ythagorean tripets. The coprime ythagorean tripes of the formuae can be used to generate and test a whoe host of infinite possibe prime numbers. REFERENCES [1] Boston N. The proof of Fermat s ast theorem. University of Wisconsin-Madison, 003. [] Darmon H, Diamond F, Tayor R. Fermat s ast theorem. Curr Dev Math, 1995, 1: 157. [3] Darmon H, Diamond F, Tayor R. Fermat s ast theorem. Eiptic curves, moduar forms and Fermat s ast theorem. Int ress, Cambridge, MA, USA, [] Singh S. Fermat's ast theorem: The story of a ridde that confounded the word's greatest minds for 358 years. Fourth Estate, [5] Edwards, Harod M. Fermat's ast theorem: A genetic introduction to agebraic number theory. Springer Science & Business Media, [6] Weisstein EW. Fermat's Last Theorem. Wofram Mathword, 00. [7] Wies A. Moduar eiptic curves and Fermat's ast theorem. Ann Math, 199, 11: eagia Research Library 6

(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].

(f) is called a nearly holomorphic modular form of weight k + 2r as in [5]. PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform