Dr. Nilaish’s Journal for Mathematics. ISSN 0974 – 3022 . An International Electronic Journal for Pure & Applied Mathematics. Journal of the Nilaish Mathematical Society On – line ( NMS ON – LINE ) ., Vol.2, N0. 21, Aug. 2008. Web. : http://nilaish.livejournal.com/ , © by Author 2008. Email : < [email protected] > . Khagaria, Bihar, India.

Fermat’s Last Theorem , does it exist ? By Nilaish* “I have discovered truly a marvellous proof of this, which this margin is too narrow to contain.” P. D. Fermat ( 1637)

For my mother, father and my sister. Truth is God.

Introduction The holy grail of mathematics : FLT , seems to be shattering on the findings of some concerete facts that : the conditional demarkation about n in the below exact theorem seems to be wrong or contains some error. Let us , observe the original statement , what Fermat wrote in around 1637 in his copy of Diophantus’ Arithmetica [1] in latin after reading[2] , “… given a number which is square , write it as a sum of two other squares”, then Fermat wrote in the margin, “Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.” In English, “ It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.” 1

Thus, in 1670 world coined this conjecture , and named it what we call it today, Fermat’s Last Theorem , . We will denote this theorem by using ( FLT )n . Many attempts were made to prove (FLT)n , but the final credit goes to English Mathematician , Andrew John Wiles[3]. We should not forget other mathematicians who lightened our way towards proof of this mighty everlasting problem. Among them who contributed to it partly and made significant pathways to its proof, are, Fermat himself proved (FLT)4 [4] , Karl Gauss proved (FLT)3[5] , Peter Dirichlet and Andrian Legendre proved (FLT)5 [6], Ernst Kummer began studying the ideal class group of Q ( ζ ) , which is a finite group that measures how far Z [ ζ ] is from being UFD [7], Also Kummer defined in his papers , regular primes and proved , where h ( Q ( ζ ) ) denotes the order of the ideal class group. In 1823, Sophie Germain found a simple proof that l is a prime with 2l + 1 prime then the first case of (FLT)l is true. Arthur Wieferich proved in 1909 that if l is a prime with , 2l - 1 ≢ 1 ( mod l 2 ) , then the first case of (FLT)l is true. This proof fails scarcely except only known examples are , 1093 and 3511. Moreover , similar conditions are known if , p l – 1 ≢ 1 ( mod l 2 ) and p is any prime ≤ 89 [8]. This helps us to prove the first case of (FLT)l for many l . Before, Wiles , (FLT)l was known for all primes 2 < l < 4 X 106 [9] . This was to check that the conjecture of Vandiver ( actually originated from Kummer and his definitions were redifined ). That l ∤ h ( Q ( ζ + ζ -1 )) holds for these primes 2 < l < 8.7 X 1020 .

Modern Methods of proof

In 1916, Srinivasa Ramanujan, FRS proved, ∆=q Then,

2

≡ σ11 we consider ∆ all

, where, σk

= ∑d|n dk . ∆ is a modular form ; if

∆ satisfies , i.e., for all z in the upper half – plane with the case

and

( z ) . And so on. Andre’

Weil and John Tate in 1940 and 1950 respectively, did the study of elliptic curves, that the curves of the form, This led to the Galois representation , that is continuous homorphisms / Q ) arrow to GL2( R ) , where R is a complete local ring such as the finite field, Fl or the ring of l – adic integers Zl . So, in particular , given elliptic curve E is defined over Q ( which means coefficient of Q. Any rational prime l , associated Galois representations ,

/ Q ) arrow to GL2 ( Zl ) and by

reduction mod l , Ǿl, E = / Q ) arrow to GL2( Fl ) . This encodes much about the elliptic curves. Likewise many mathematicians , like Jean – Pierre Serre[10], Pierre Deligne[11] in 1969, for weights k > 2. For k = 2 we have work of Martin Eicher and Goro Shimura[12] . For k = 1 , later Deligne and Serre[13] established the requsite. The representations of share many similarities with the . Summing it up we have a conjecture of Yutaka Taniyama of 1955, which was given finishing touch by Shimura, which would attach a modular form of this kind to each elliptic curve over Q. Therefore, we have ,

{ Rep. from elliptic curves } | { Rep. from certain modular forms } subset of { Admissible Galois Rep. }

In 1985, Gerhard Frey gave us a link to proof of FLT, If we consider Frey curve, we have clearly, phi bar representations have properties that a representation associated with modular form should not. 3

Andrew J. Wiles[14] approach rests on two theorems, the proof is extremely simple to understand, since it follows from just 2 theorems : Theorem 1: If there is a solution (x, y, z, n) to the Fermat equation, then the elliptic curve defined by the equation, is semistable but not modular. And Theorem 2: All semistable elliptic curves with rational coefficients are modular. The proof rests on the results of Ribet (proving Theorem 1) and Wiles (proving Theorem 2). Together they proved FLT. You can read this first if you just want the highest-level overview of the proof. However, both of these theorems are very difficult themselves, and both have been proven only in the last 10 years. But given that both are now known, it follows that, in order to avoid a contradiction, there cannot be any solution to the Fermat equation. Just oppositeof which I’ll try to show later in this Paper. The flaw in the Wiles’ paper presented in 1993 at Isaac Newton Institute in Cambridge, UK, and in the final paper of 1994, was very well pointed out by Gerd Faltings [15], Which says, “ The proof of FLT ( conjecture ) was finally completed in september of 1994. A. Wiles announced this result in the summer of 1993 ; however , there was a gap in his work. The paper of Taylor and Wiles doesnot close this gap but circumvents it.” My one problem left for solution[16] accompanies , Perelman(The man who turned down the prestigious Field's Medal and one million dollars after proving the famous Poincaré's Conjecture, the Russian mathematician Grigori Perelman), made headlines today in the small North-Siberian village of Ralp-Olifo-Chuk. has been investigating the tiny (but often downplayed) discrepancy between the numbers 0.999... and 1, in relation to Andrew Wiles' corrected proof of Fermat's Last Theorem from 1994. Excitingly, Perelman has shown that Fermat was wrong after all. For the main tool of his proof, Perelman constructed a new number class - unreal numbers 4

furthering the path paved by imaginary and surreal numbers. The output from unreal calculations is decided by polarising light through a quantum computer. These new PC's, equipped with the new Microsoft Quantista operating system, work to an amazing calculational accuracy, but unfortunately their output cannot be read: The reading process itself might change the delicate output values. However, Perelman has used String Theory, Membrane Theory and the new Pampers Nappy Theory to establish probability waves for unreal equations. For example, the value of 1+4+2008 behaves according to the normalised horseshoe distribution, (as long as the Ricci tensor star signs are in a suitable orientation). According to one related theorem, all things look like an egg, if looked at closely enough. Perelman is clearly disturbed by the skepticism of fellow mathematician Shing – Tung Yau, who expressed his concerns after it turned out that Perelman's proof was written in a remotely recognised dialect of Urdu, using invisible ink. "If my colleagues, Cao and Zhu, had found exactly the same proof, I would have no doubts about its consistency," Zau exclaimed. Reuters managed to locate a Gandalf-bearded Grisha. Any ways, what he says is not very well published. So, Here I am trying to point out cases regarding my heading.

Why it seems that it is so. Hence, it is clear from above two theorems that , Proof supplied by Wiles rests on these theorems. I’ll like to put few observations on the some aspects of the proof, i.e., Elliptic Curves , Modular forms and Deformations . Elliptic Curves Usually, according to Wiles E is defined over the rational numbers Q. Which means the coefficient of f are in Q. As pointed out by Gerd Faltings that all three zeros of f are distinct ( E is non singular ). We may consider E as those solutions in Q, R, or C, denoted respectively , E ( Q ), E ( R ) and E ( C ) . One usually includes in this set an infinitely distant point, denoted by ∞ . The solution set has the structure of an abelian group , with ∞ as the neutral element. The inverse of ( x, y ) is ( x , - y ), as we sum of three 5

points vanishes if they lie on a line. The group addition is given by algebraic functions. As deploying Mordell’s Theorem , group E ( Q ) is finitely generated, E ( R ) is isomorphic to R/Z or to R/Z X Z/2Z, and E ( C ) C/ lattice ( for example, yields the lattice Z direct sum Zi ). Further , if we let integer n let E [ n ] denote the n division points. C is also isomorphic and so on. As we have obtained above Galois Represenations based on this principle. If we change the coordinates , then we see f has integral coefficients , after manupulations. Likewise , if one then reduces modulo a prime number m , he obtains a polynomial over the finite field Fm . If the zeros of that reduced polynomial are distinct then it gives an elliptic curve over Fm . And , this holds good for all primes m, except for the finitely divisors of the discriminant of f . Once again the choice of f is not unique, but we say E has good reduction at m , if we find an f such that the zeros modulo m are distinct. This observation is not thoroughly , true at m = 2 because the term , otherwise E has bad reduction at m . In this case only two zeros of f modulo m coincides, one says E has semistable bad reduction. E is called semistable if all m it has either good or semistable reduction. The curve , , is not semistable at m = 2. An example of a semistable is Frey curve. A solution of FLT , , The curve so found is,

This curve has bad reduction at exactly at the prime divisors of

. Now this has

very remarkable property , i.e., consider the linked Galois representation , / Q ) arrow to GL2( Fl ) . This representation is not supportive at all prime numbers m at which E has good reduction. If m = 1, then it has non – supportive nature or crystalline nature. The particular form of the equation for E , this is also true at all prime divisors m > 2 of . Whence, l – division points behave as if E had good reduction at all m > 2. Moreover, we see , there are no semistable elliptic curves over Q with this property, hence, derives contradiction . To resolve it we need to remove elliptic curves by Modular forms. Modular forms 6

Again by first definition E is semistable but not modular, according to Wiles. But by the second definition , all semistable elliptic curves with rational coefficients are modular. Above it is shown that there is no semistable elliptic curves over Q. Which again contradicts his proof on this ground. Also, it may be possible to show that E is modular iff the associated l – adic representation can be constructed in this manner.

Deformations As shown by Faltings , it was only possible to show that p is annihilated by η. This is the content of the theorem given by M. Flach. The higher levels of the Euler system, could not be constructed. Summing comments on Wiles’ proof He got assistance from Nick Kartz, Ribet, Taylor, and cropped the minimal case by, assuming above two theorems mentioned. To reduce minimal case , he has to estimate how both sides of the inequality, #p/

≥ #Z3/η . Z3 .

Change as one can proceed from level M to higher level N ( M | N ). And , principles stated by Ribet and Ihara by luck agrees on the considered condition by Wiles. Which accomplished his counter example proof. A few remarkable equations. Why Fermat never published or wrote his proof of this conjecture? I’ll try to mention few new discoveries by other mathematicians, Marietta Georgia[17], I. Savant[18] of Georgia. Mr. Savant says, “I think Fermat succumbed to pressure when he claimed that he had found a proof, and I don't blame him. I mean, there's this theorem named after you, and they even tell you that it's the last one you're getting. Hell yeah, you're going to tell them you proved it. For years people have tried to show that Fermat's Last Theorem is true. Some have tried to show it was untrue, and others have tried to show that it was not-not-not un false . It dawned upon me that no one had really tried to show that 7

it was un-not not-not-anti-not untrue. When I looked at it this way, I immediately found that it was what I just said it was, and at that point I knew I had stumbled upon a great discovery”. When he was reminded about recently published proof by A. Wiles and R. Taylor, he says, “Yes, I saw that proof, and it was a valiant attempt at a futile endeavor. The gentleman that published it obviously is pretty good, but I didn't like the way he kept prefacing every paragraph with 'This isn't going to make much sense, but trust me on it.' And I really thought that the pathetic plea for money to investigate Fermat's Second to Last Theorem was very unprofessional. But I did like the way he presented most of it in stick – figure cartoon form. That was neat”. Savant says that this problem is unsolvable, when we take aid of traditional theory. Without his own expertise in Number Addition and Subtraction Theory, advanced superscripting, and home brewing techniques, it would have been impossible. Using an innovative combination of the theory of general relativity, quantum mechanics, and his own findings in the field of Jello Mold Topology, Savant first showed that any given number has a high probability of being equal to itself. This leads to the observation that there is actually a small probability that the number is not equal to itself, but rather a different number entirely. He defined a variable "idio" to be the probability that a given number is equal to another given number. He illustrates his next step thusly: “Obviously, the probability that, say, the number 10 is equal to 11 is very very small, although it is thought to have happened in June of 1952. But remember, quantum mechanics is screwy, and the obvious is sometimes unapparent. The probability that the number 10 is actually 10.5 is alot higher than the probability that it is 11, and as we approach 10 itself, the probability gets higher and higher. The idio becomes significant”. Savant showed that for an arbitrary band of values centered around a given number, there is a corresponding range of distinct probabilities, which he called a "Bunch of Idios." He notes : “I knew as soon as I had stumbled onto distinct probabilities that we were talking real possibilities. That means that for a given bunch of idios, there is a corresponding range of values that the number itself falls into. Savant continues: I 8

saw that this discovery could lead to a powerful new algorithm: pick a number, any number, and determine what it might be equal to”. Savant calls this breakthrough the "Idiotheorem," and he knew immediately that it was the key to disproving Fermat. Mr. Savant thinks that the Idiotheorem is almost as good as his Theory of Bad Astronomical Perspective, which says, in simple terms, that in some corners of the Universe there are patterns of stars that exactly resemble very intricate objects like bicycles, microwave ovens, and Barney Fife, but we can't see anything that neat because the Earth has a very bad seat when it comes to constellation viewing. The immediate task at hand was to assemble a sophisticated collection of computational equipment. This perhaps, ironically, was the most difficult part of Savant's historical feat. “I knew that a monochrome monitor would be essential. The slightest hint of color might throw the calculations way off. And I knew that I couldn't use anything built after 1991, the last recorded palindromic year, which is evenly divisible only by 11 and 181, both palindromes themselves. A 286 wouldn't be broken in enough, so I chose a 285”. And then the magic began. “It took a whole day to set up the program. A mysterious bug kept making it lapse into a primitive form of WordPerfect. I was tempted to halt the entire project, because I really liked the simple, intuitive text editing features, but I pressed on anyway. I first picked my group of idios to be a band of +/- 1% variance. I was amazed to find that there were hundreds of solutions filling the screen, for any power you could possibly think of. But that was too easy; it was just a warmup. I narrowed the idios to +/- 0.01%, and I still got dozens of solutions. I knew that my next step would finally put the Fermat matter to rest for all of eternity. Nothing would have been proved if I didn't get the idios down to actual quantum levels. I set the value to 0.0001%. This is similar to saying that the number 1 will fall into a range of values that differ from 1 by less than 1 millionth. That's a no brainer. I knew that any solution that met this criteria would surely be as completely true as the very Idiotheorem that I based it on”. Success came at one stroke before midnight. The magic and power of the Idiotheorem yielded several solutions to the Fermat Equation. Fermat had finally 9

been proved wrong, and the tortured souls of a thousand mathematicians finally found rest. A formal presentation of Savant's findings is in the works. But for the skeptic and believer alike, Savant allowed this reporter to publish the very first Fermation Solution in all the history of mathematics. Here it is:

And a few more that followed:

Solutions were also found for the 4th, 5th, 6th, and 7th powers. These solutions are all accurate to within 1 millionth, well within the range of probabilities defined by the Idiotheorem. Hence, we see the claims of Mr. Savant. Also for n = 12, we have,

As in my paper[19], I have clearly, raised the same issue, taking in consideration of above two equations. As if traditionly we can assume , 0.999… = 1. According to the theory of Mr. Savant. Again FLT is in wrong box. In my paper on [20], I have clearly found that, for all even numbers we have x, y, z positive integers. No matter n > 2. A few new analysis’ on FLT. This was faired by me using complex analysis, my theory on solution of Diophantine equation[21], to test the consistency of FLT. Here we will witness it, Complex Analysis From the first proposition we have, 10

setting ,

, hence , we get,

Or, we can write,

, using complex numbers, we have,

, where . As we know , Z is subset of C, therefore, ( 1 + ki ) is also pure complex number. i.e., Im( Z ) not zero. Whence, the nth root of the expression is given by, +

] . where, s = 0, 1, 2, … , ( n –

1 ). Which means, is not zero. If possible let it be zero then after further working we will find one case in which k = 0. But for other values of s , k is not equal to zero. Which again makes FLT a conjecture, and makes it highly unsolvable. Number Theory Analysis Let the (FLT)n be correct. Then we have Now, we have , A + B = 1, As we all know this linear Diophantine equation has infinitely many solutions , apart from Bezout’s Identity[22]. The solutions can be given by, A = h and B = ( -h + 1 ) and vice – versa , where, h = 0,1,2,3,&C.

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But , by using my theorem 3.

Theorem 3 “For every equation of type ax + by = c , the supplementary equation is ax – by = 1/c”. ( where all symbols have usual meaning ). Proof of the theorem 3 can be found in my paper on linear Diophantine equation. Note that for above equation when we apply this theorem, we get A + B = 1 and A – B = 1. When we solve it, we get, A = 1 , B = 0 and vice – versa . Which means , x = z and y = 0, taking z not zero. Again vice – versa . Which casts serious doubts on FLT, as x, z can be positive integers and likewise others cited in vice – versa condition. The very question of validity of theorem 3, I would like to remind that, this theorem can be used to find solutions of linear Diophantine equation s , ax + by = c. There is also one open question , i.e., when we devise formulae out of ax + by = c and using theorem 3, we get general solution to all equations of this family, i.e., . But, it can’t be used directly for equations like x + y = 1. The reason behind this is, when c = 1 , we have directly from the formulae, x = 1/a and y = 0, which makes my theorem doubtful as per its treatises we can’t use it for equations , ax + by = 1 , rather we have to use the theorem 3 directly itself for working. Some other approaches We can have cases from equations of X and Y. X and Y are complex numbers for 2. X = Y , which cannot be possible. 3. X and Y be some rational numbers , which lie in 1.

Case 1, says X or Y is a complex number. Which, makes FLT wrong. Case 2, doesn’t exist. Case 3 , needs working to arrive at some conclusion. Let us observe the steps. 12

, Now, by using Binomial Theorem for any index we have, . =1+

, for FLT to be right,

= 0. Which is

again problematic and there is no way to arrive at the required result. Trigonometry Also taking equation ,

Let us suppose a right angle triangle,

with inclination θ. Then this equation transforms if (x/z) and (y/z) are Trigonometric ratios Sin and Cos respectively. We get resulting equation , Sin n θ + Cos n θ = 1. And we all know that , n = 2 for all cases is valid for this identity. Which proves FLT conditionally.

Conclusion As set forth in different sections of this paper it is clear that ( FLT )n , creates serious doubts on very nature of the fixed variable n. All semistable elliptic curves are not modular, but it shown that it is may be modular. Perelman raises his veto on addressing to the press. Mathematicians from Georgia , showed few equations under Idio-variance theorem . My treatments clears the illusion very well. As taking my findings and other discoveries in mind one can say that , “ Fermat’s Last Theorem, does it exist?”

References [1] Arithmetica , Diophantus, On a copy of this book, Fermat wrote, this famous problem, But, the book was lost. [2] Problem No. 8, Arithmetica, Diophantus.

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[3] Chair , Deptt. Of Mathematics, Princeton University, USA. [4] Nigel Boston, The Proof of Fermat’s Last Theorem, University of Wisconsin, Madison, 2008, p. 6. [5] K. Ireland and M. Rosen. A classical introduction to modern number theory. Second edition. Graduate texts in mathematics 84. Springer – Verlag , New York, 1900. p. 285, 43. [6] Nigel Boston, The Proof of Fermat’s Last Theorem, University of Wisconsin, Madison, 2008, p. 10. [7] I. Stewart and D. Tall. Algebraic Number Theory and Fermat’s Last Theorem. Third edition. A K Peters , Ltd. , Natick, MA, 2002. [8] A. Granville and M. Monagan. The first case of Fermat’s last theorem is true for all prime exponents up to 714, 591, 416, 091, 389. Trans. AMS, 306 : 329 – 359 , 1988. [9] J. Buhler , R. Crandall , R. Ernvall and T. Metsankyla. Irregular primes and cyclotomic invariants to four million. Math. Comp. , 61, no.: 203 : 151 – 153 , 1993. [10] Nigel Boston, The Proof of Fermat’s Last Theorem, University of Wisconsin, Madison, 2008, p. 13. [11] P. Deligne . Formes modulaires et representations l – adiques . Séminaire Bourbaki, p. 139 – 172 , 1971. [12] G. Shimura. Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1971. [13] P. Deligne and J. P. Serre. Formes modulaires de poids 1. Annales de l’Ecole Normale Superieure, 7: 507 – 530 , 1974. [14] Paper of Andrew John Wiles in Annals of Mathematics, 1994 – 95 . [15] Gerd Faltings , The proof of Fermat’s Last Theorem by A. Wiles and R. Taylor, Notices of the AMS, Vol. 42, No.7, p. 743 – 746 , 1995.

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[16] , [19] Nilaish, Is 0.999… = 1? , Dr. Nilaish’s Journal for Mathematics, ISSN 0974 – 3022 , Vol. 1, No. 19, p. 38, 2008. [17] , [18] IP Atlanta , The mathematical community, press release 1995, Twisted Ruler Productions, Email : [email protected] . [20] Nilaish, Two equations about FLT , Dr. Nilaish’s Journal for Mathematics, ISSN 0974 – 3022 , Vol. 1, No. 18, p. 36 – 37 , 2008. [21] Nilaish, On the solution of the linear Diophantine equation, Dr. Nilaish’s Journal for Mathematics, ISSN 0974 – 3022 , Vol. 1, No. 2, p. 6 – 9 , 2008. [22] Jones and Jones, Elementary Number Theory, Spriger – Verlag , India, ISBN 81 – 8128 – 278 – 7 , chap. 1, 2008. This paper is devoted to my parents, sister and truth. * Member of EWDM of IMU, Germany, IMS ( INDIA). Author and referee of Journal of Royal Society, London, UK. For more details please log on to my : HOMEPAGE : http://sites.google.com/site/wesiteofnilaish/ EMAIL : [email protected] Permanent Address : S/O Dr. S. N. Sharma, HOD, History, Koshi College, Khagaria, Bihar – 851205 , India.

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