Fermat's Last Theorem

Fermat's accidental challenge " ...nullam in infinitum ultra quadratum po-testatem in duos eiusdem nominis fas est dividere " , which means " it is impossible to separate any power higher than the second into two like powers ". That is in modern mathematical jargon the equation x n + y n = z n has no solution in natural numbers for n greater that two. And then it continues: " cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet " , which means: " I have discovered a truly marvellous proof of this, which this margin is too narrow to contain ". This remark was written in the year 1637 by the French lawyer and amateur mathematician Pierre de Fermat in the margin of his copy of Diophantus' Arithmetica. He certainly didn't expect that it would keep mathematicians, professionals and amateurs alike, busy for centuries trying to unearth the proof. From the innumerable things we've learnt in school maths the Pythagorean Theorem is one that remains with us more anything. As we leave school and for many years after, when we thing about Pythagorean Theorem, we immediately draw a right triangle in our minds eye. A triangle that has a ninety-degree angle. Most will remember that the hypotenuse is the side of the triangle directly across from the right angle. It is the only side of the triangle that is not a part of the right angle. And we learned the relationship between the sides of a right triangle (see fig 1.) When c stands for the hypotenuse and a and b are the sides forming the right angle.

If Fermat’s Last Theorem were false, this would require either a conspiracy theory , or a quasi-conspiracy theory.

“The conspiracy theory, of course, would be that mathematicians as a body know that Fermat’s Last Theorem is false, but do not want everyone else to know this, so they claim that they have verified the proof and found it valid, while in reality there are flaws in it and they know about them.

The quasi-conspiracy theory would be that mathematicians as a body believe that Fermat’s Last Theorem is true, but that they consistently fail in their attempt to verify the proof. There is a mistake in it, but each time someone tries to verify it, they fail to notice the mistake.

The reason to call this a quasi-conspiracy theory is that the most reasonable way for this to happen is if mathematicians as a body have motivations similar to the mathematicians in the case of the actual conspiracy, motivations that cause them to behave in much the same ways in practice.

We can see this by considering a case where you would have an actual conspiracy. Suppose a seven year old child is told by his parents that Santa Claus is the one who brings presents on Christmas Eve. The child believes them. When he speaks with his playmates, they tell him the same thing. If he notices something odd, his parents explain it away. He asks other adults about it, and they say the same thing.

The adults as a body are deceiving the child about the fact that Santa Claus does not exist, and they are doing this by means of an actual conspiracy. They know there is no Santa Claus, but they are working together to ensure that the child believes that there is one.

What is necessary for this to happen? It is necessary that the adults have a motive quite remote from truth for wishing the child to believe that there is a Santa Claus, and it is on account of this motive that they engage in the conspiracy.

In a similar way, suppose that mathematicians as a body were deluded about Fermat’s Last Theorem. Since they are actually deluded, there is no actual conspiracy. But how did this happen? Why do they all make mistakes when they try to verify the theorem? In principle it might simply be that the question is very hard, and there is a mistake that is extremely difficult to notice. And in reality, this may be the only likely way for this to happen in the case of mathematics. But in other cases, there may be a more plausible mechanism to generate consistent mistakes, and this is wishful thinking of one kind or another. If mathematicians as a body want Fermat’s Last Theorem to be true and to be a settled question, they may carelessly overlook mistakes in the proof, in order to say that it is true. Technically they are not making a deliberate mistake. But in practice it is the lack of care about truth, and the interest in something opposed to truth, which makes them act as a body to deceive others, just as an actual conspiracy does.”

The fact is, of course, that the proof of the Wiles is not widely accepted and understood, therefore it is a proof of conspiracy .

However, the Last theorem of fermat as a theorem is correct, the healthy mathematical community expects a proof accessible at its average spiritual level so that it is a clear proof and within the real mathematical theories.

The purpose of this paper is to provide a proof that is simple , understandableand and widely accepted. Any critique is acceptable. Thanks for the scientific world of academia.edu.

This paper proves Fermat's Last Theorem (FLT) using 2-d plane geometry and algebra. Proof is based on a "nested, generalized, and averaged Pythagorean diagram system (NGAPDS, Fig. 2) drawn within a 2-d coordinate system on standard square gridlines, and analogous derived graphs. In the given equation of FLT, natural numbers x,y are held constant while exponent n is allowed to vary smoothly through all positive reals as an independent variable. Dependent values of z then vary inversely with n (e.g., Fig. 3). Analyzing the NGAPDS, which features geometrically equivalent square areas at all scales (Fig. 4), we derive two inversely related curves simultaneously describing z in an analogous 2-d (n, z) coordinate system (e.g., curves labeled A and B in Fig. 8). These simultaneous curves feature a unique common solution point on the n = 2 axis of the derived n, z coordinate system, this point representing excluded maximum due to the inverse relation between n and z. Because the only solution z value in this stipulated diagram system, for any given values of x,y we have chosen as constants, arises at the excluded maximum of the NGAPDS, we arrive at a fully generalized and comprehensive proof by contradiction for FLT.

If we simulate the 2 equations, it turns out that when n> 2 for c^n + b^n = a^n if n = 2k + 1 , k> = 1 in Z +, only 2 things can happen 1. (sinx = 0 and cosx = 1) or ( sinx = 1 and cosx = 0) 2nd. if n = 2k, (cosx = + / - 1 and sinx = 0) or ( sinx = + / - 1 and сosx = 0) will hold, where k> 1 in Z +. Otherwise  (sinx) ^ n + (cosx) ^ n <1 (n = 2k or n = 2k + 1) and therefore equality is impossible .Only in the case n = 2 is equal to 1 as a maximum i.e. (sinx) ^ n + (cosx) ^ n = 1. On the logic that if the limiting case = 1 arises then the results found in the paper are obtained.Οtherwise obviously we will have a form of inequality so the equality will not hold and hence the solution for n>2 is falsified..This is a simulation method since we assume that the angle for exponent n>2 is approximately equal to exponent n=2..Τhis paper is a supplement to the 2nd paper found on the website,https://www.academia.edu/39519597/FERMAT_LAST_THEOREM_PROOF

Here an attempt is made to find positive numbers a,b and c such that a^n+b^n=c^n. Two cases were considered. a^n+b^n is not divisible by (a+b)^2 and a^n+b^n is divisible by (a+b)^2. From this a solution for the case n=3 is obtained as (9+5)3 + (9^3-sqrt(5))^3 = 12^3. A condition for finding such numbers for any n is also reached using elementary number theory in 3 pages.

All eyes are on the Riemann's hypothesis, zeta, and L-functions, which are false, read this paper. The Euler product converges absolutely over the whole complex plane. Using factorization method, we can prove that Riemamn's hypothesis and conjecture of Birch and Swinnerton-Dyer are false. All zero computations are false, accurate to six decimal places. Riemann's zeta functions and L  functions are useless and false mathematical tools. Using it one cannot prove any problems in number theory. Euler totient function () n  and Jiang's function 1 () n J   will replace zeta L  and functions.

The distinction between the Domain of Natural Numbers and the Domain of Line gets highlighted. This division provides the new perception to the Fermat's Conjecture, where to place it and how to prove it. The reasons why the Fermat's Conjecture remained unproven for 382 years are examined. The Fermat's Conjecture receives the proof in the Domain of Natural Numbers only. The equation a n + b n = c n with positive integers a, b, c, n is not the Fermat's Conjecture in the Domain of Line.

Fermat’s Last Theorem, now Fermat-Matos’ Theorem

Part I : odd n

Whichever the whole numbers x, y, z (relative primes) and n may be,
such that the sum x^n + y^n equals z^n, there is one whole number and one only ― denominated z’― that equals the sum x + y, which ― apodictically ― divides z^n; and thus, e.g., z’- x, alias y, also divides exactly z^n - x^n, alias y^n.

Now, the remainder of such exact division (the quotient of which is y^n-1) is necessarily zero, what implies that z’^n = z^n, therefore z’= z (the appendix turns this utmost important point crystal clear).

Consequently, since (x + y)^n results equal to x^n + y^n, it follows that n - 1 (odd n) is zero, and hence ― Q.E.D. ― n = 1.

"In this age in which mathematicians are supposed
to bring their research into the classroom, even at
the most elementary level, it is rare that we can turn
the tables and use our elementary teaching to help
in our research. However, in giving a proof of Fermat’s
Last Theorem, it turns out that we can use
tools from calculus and linear algebra only."

Titled in Portuguese "Memoir from the House of the Dead", this is a document attesting the absolutely extraordinary sepulchral silence that a national scientific academy ─ named just after the country's capital: the Lisbon Academy of Sciences ─ has had the grotesque flair to decidedly dispense to a citizen who candidly knocked at their door asking for the review of a mathematical achievement of his as significant as the proof of Fermat's Last Theorem… the first time in 1986.
The Anglophonic readers may find below a particular note at their convenience.

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics. A simple relation of square numbers, which encapsulates all the glory of mathematical science, is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20 th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be measured as integers and whose surface can be square. This problem was solved negatively by Fermat in the 17 th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. The difficulty of solving Fermat's equation was first circumvented by Willes and R. Taylor in late 1994 ([1],[2],[3],[4]) and published in Taylor and Willes (1995) and Willes (1995). We present the proof of Fermat's last theorem and other accompanying theorems in 4 different independent ways. For each of the methods we consider, we use the Pythagorean theorem as a basic principle and also the fact that the proof of the first degree Pythagorean triad is absolutely elementary and useful. The proof of Fermat's last theorem marks the end of a mathematical era; however, the urgent need for a more educational proof seems to be necessary for undergraduates and students in general. Euler's method and Willes' proof is still a method that does not exclude other equivalent methods. The principle, of course, is the Pythagorean theorem and the Pythagorean triads, which form the basis of all proofs and are also the main way of proving the Pythagorean theorem in an understandable way. Other forms of proofs we will do will show the dependence of the variables on each other. For a proof of Fermat's theorem without the dependence of the variables cannot be correct and will therefore give undefined and inconclusive results.

A proof of Fermat's last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat's last theorem in the case of í µí±›í µí±› = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat's approach of infinite descent. The infinite descent is linked to induction starting from í µí±›í µí±› = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat's last theorem. As far as Fermat had been proved the theorem for í µí±›í µí±› = 4, one can suggest that the proof for í µí±›í µí±› ≥ 4 was accessible to him.

Non-computable functions can seem so quixotically remote from everyday mathematics that we may imagine they have little relevance to more familiar domains of application. That this isn't true will be demonstrated here using the proofs set forward by Tibor Rado in his 1962 article in The Bell System Technical Journal -On Non-computable functions.

New E-Book “In Depth” Reveals Hitherto Unknown Formula to Apply Pythagoras
Theorem in Three Dimensions

(Tbilisi, Georgia, September 26, 2017) – Amiran Kapanadzes’ new e-book, "In Depth", is an insightful and thought provoking look at the application of Pythagoras theorem to three dimensional space. "In Depth" goes on to show why the Pythagoras theorem does not work in three dimensions, and what is the regularity of hypotenuse and legs development in cube with new formula.
"In Depth" is all about the workings of the human mind, and pushes against the limits of rationality and the fiction of determinism. Available at all leading online publications, "In Depth" reveals a new equation for right angled triangle progression in three dimensions. The proposed new solution is bound to attract renewed attention to this aspect of geometry and math which is well known even to school students.
"In Depth" contains not only a new proposed solution, but also describes author’s own personal experiences and the creative process he used to obtain the solution for square progression in 3D space and the new equation for right angled triangles
“All the scientific discoveries are the result of searching rationalism in once chaotic and undetermined mental space. Real harmony can be achieved with our precious mind if we'll understand the nature of metaphysics,” says Amiran.
Amiran Kapanadzes holds a bachelors’ degree in business administrating from Georgian Agrarian University and currently studies MA program of Political Science at the I. Javakhishvili Named Tbilisi State University. He has also been the International Department Head at the NGO Economic Policy Experts Center.
---------------------------------------------------------------------------------------------------------------------
For more information, please visit: http://www.lulu.com/shop/amiran-kapanadze/in-depth/ebook/product-23330464.html

Hallar un algoritmo para determinar si, dada una ecuación
polinómica diofantina (1) con coeficientes en los enteros, por ejemplo:
Ax + By + Cz + Dw = K (1)

Tiene o no solución en los enteros, este problema fue resuelto en 1970
por el matemático ruso Yuri Matiyasevic.
Su respuesta tan impresionante como la de Galois, fue que no existe
dicho algoritmo.

Se muestra vía restos cuadráticos que no hay soluciones enteras
para la curva $E_{1}$ cuando $K= (4n - 1)^{3} - 4m^{2}$  donde $m$
y $n$ $\in {\mathbb{Z}}$ tales que ningún primo $ q \equiv -1 (
mod  4 ) $ divide a m . Y se hallan las soluciones racionales para
la curva $E_{2}$.

This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists and mathematicians. Precisely, we have described some mathematical results regarding the Fermat's Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan's modular equations, how primes and adeles are related to the Riemann zeta functions and the p-adic and adelic string theory. Furthermore, we show that also the fundamental relationship concerning the Palumbo-Nardelli model (a general relationship that links bosonic string action and superstring action, i.e. bosonic and fermionic strings in all natural systems), can be related with some equations regarding the p-adic (adelic) string sector. Thence, in conclusion, we have described some new interesting connections that are been obtained between String Theory and Number Theory, with regard the arguments above mentioned. In the Chapters 1 and 2, we have described the mathematics concerning the Fermat's Last Theorem, precisely the Wiles approach in the Chapter 1 and further mathematical aspects concerning the Fermat's Last Theorem, precisely the modular forms, the Euler products, the Shimura map and the automorphic L-functions in the Chapter 2. Furthermore. In this chapter, we have described also some mathematical applications of the Mellin transform, only mentioned in the Chapter 1, the zeta-function quantum field theory and the quantum L-functions. In the Chapter 3, we have described how primes and adeles are related to the Riemann zeta function, precisely the Connes approach. In the Chapter 4, we have described the p-adic and adelic strings, precisely the open and closed p-adic strings, the adelic strings, the solitonic q-branes of padic string theory and the open and closed scalar zeta strings. In the Chapter 5, we have described some correlations obtained between some solutions in string theory, Riemann zeta function and Palumbo-Nardelli model. Precisely, we have showed the cosmological solutions from the D3/D7 system, the classification and stability of cosmological solutions, the solution applied to ten dimensional IIB supergravity, the connections with some equations concerning the Riemann zeta function, the Palumbo-Nardelli model and the Ramanujan's identities. Furthermore, we have described the interactions between intersecting D-branes and the general action and equations of motion for a probe D3-brane moving through a type IIB supergravity background. Finally, in the Chapter 6, we have showed the connections between the equations of the various chapters.

UPDATED AND REVISITED VERSION - 05.11.2021

https://arxiv.org/abs/1703.10132
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with number theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For the derived polyadic ring operations neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem for the Tarry-Escott problem is presented.

This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists and mathematicians. Precisely, we have described some mathematical results regarding the Fermat's Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan's modular equations, how primes and adeles are related to the Riemann zeta functions and the p-adic and adelic string theory. Furthermore, we show that also the fundamental relationship concerning the Palumbo-Nardelli model (a general relationship that links bosonic string action and superstring action, i.e. bosonic and fermionic strings in all natural systems), can be related with some equations regarding the p-adic (adelic) string sector. Thence, in conclusion, we have described some new interesting connections that are been obtained between String Theory and Number Theory, with regard the arguments above mentioned. In the Chapters 1 and 2, we have described the mathematics concerning the Fermat's Last Theorem, precisely the Wiles approach in the Chapter 1 and further mathematical aspects concerning the Fermat's Last Theorem, precisely the modular forms, the Euler products, the Shimura map and the automorphic L-functions in the Chapter 2. Furthermore. In this chapter, we have described also some mathematical applications of the Mellin transform, only mentioned in the Chapter 1, the zeta-function quantum field theory and the quantum L-functions. In the Chapter 3, we have described how primes and adeles are related to the Riemann zeta function, precisely the Connes approach. In the Chapter 4, we have described the p-adic and adelic strings, precisely the open and closed p-adic strings, the adelic strings, the solitonic q-branes of padic string theory and the open and closed scalar zeta strings. In the Chapter 5, we have described some correlations obtained between some solutions in string theory, Riemann zeta function and Palumbo-Nardelli model. Precisely, we have showed the cosmological solutions from the D3/D7 system, the classification and stability of cosmological solutions, the solution applied to ten dimensional IIB supergravity, the connections with some equations concerning the Riemann zeta function, the Palumbo-Nardelli model and the Ramanujan's identities. Furthermore, we have described the interactions between intersecting D-branes and the general action and equations of motion for a probe D3-brane moving through a type IIB supergravity background. Finally, in the Chapter 6, we have showed the connections between the equations of the various chapters.

UPDATED AND REVISITED VERSION - 22.10.2021

Der Große Satz von Fermat, und die Additionskonstanten von Potenzen größer als 2. dass es für die Formel von Pythagoras a² + b² = c² keine ganzzahligen Lösungen in Potenzen größer als 2 geben kann. verschiedene Bemerkungen auf den Seitenrand. Bei dieser Vermutung schrieb er, dass er eine wunderschöne Lösung dafür hätte, der Rand wäre jedoch zu klein, um sie zu fassen. Meine eigenen 15-jährigen Forschungen, die am Anfang in vielen Sackgassen mündeten, haben am Ende doch einen neuen Lösungsansatz-eine wunderbar einfache Lösung : Meine Vermutung lautet :
Herr Fermat hat die Formel von Pythagoras umgebaut.
Neu Schreibweise c²-b² = a² und c^n-b^n = a^n Ob es so war können wir nie ergründen, aber mit dieser anderen Perspektive sind 5 neue Erkenntnisse entstanden. Herleitung war und ist immer c^n-b^n = a^n. O Jede Potenz hat eine (ihre eigene) Additionskonstante. O Jede Additionskonstante ist das Produkt der linearen Reihenfolge von (n) bis zur gewünschten Potenz. O Die Additionskonstante für die 4. Potenz lautet deshalb 1 x 2 x 3 x 4 = 24. Die Erkenntnisse 2 und 3 auf der Basis c^n-b^n = a^n sind linear, unendlich, und absolut, und lauten :
O Alle Einer-Zahlen-Werte Exponent = ungerade (n) haben die Eigenschaft a^n = 6n+1.
O Alle Endungen Exponent = gerade (n) haben die Eigenschaft a^n = 8n+1 und 8n-1 im Wechsel.