Fermat's Last Theorem

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup Γ 0 (N), as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equations are algebraic transformations of special functions. When N = 4, 3, 2, they are modular transformations of Ramanujan's elliptic integrals of signatures 2, 3, 4. This gives a modern interpretation to his theories of integrals to alternative bases: they are attached to certain families of elliptic curves. His anomalous theory of signature 6 turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework.

Let s = σ + Iτ , let Zeta(s) the Riemann function and η(s) the Dirichlet Eta function. They have the same roots for σ in the open interval (0, 1 2). The aim of this article is to prove algorithmically that there is no root s = σ + Iτ of Eta(s) with 0 < σ < 1 2. 1

Using the mechanical interpretation of the derivative and analyzing the function y = x^n, we can derive the general form of the equation for Newton's law of universal gravitation. From the general and more precise formulation of the law of gravitation, it follows that the gravitational constant G (if the equation is written in the traditional form) is no longer constant and will increase with increasing distance between interacting bodies.

Используя механическую трактовку производной и анализируя функцию y = x^n, можно вывести общую форму уравнения для закона всемирного тяготения Ньютона. Из общей и более точной формулировки закона тяготения следует, что гравитационная постоянная G (если уравнение записать в традиционной форме) уже не является постоянной и будет возрастать с увеличением расстояния между взаимодействующими телами.

P.S. This is the Russian version of the work: The Pioneer effect, Fermat's last theorem and the gravitational constant G.

In this work I demonstrate that a possible origin of the Frey elliptic curve derives from an appropriate use of the double equations of Diophantus-Fermat and from an isomorphism: a birational application between the double equations and an elliptic curve. From this origin I deduce a Fundamental Theorem which allows an exact reformulation of Fermat's Last Theorem. A complete proof of this Theorem, consisting of a system of homogeneous ternary quadratic Diophantine equations, is certainly possible also through methods known and discovered by Fermat,in order to solve his extraordinary equation.

In this paper, we have described some  equations concerning the functions ζ(s) and ζ(s,w) and some Ramanujan-type series for 1/π. We obtain various mathematical connections with some equations concerning the p-adic open string for the scalar tachyon field and the zeta strings.

Let s = σ + Iτ , let Eta(s) e the Dirichlet serie and Zeta(s) the Riemann function. They have the same roots for σ in the open interval (0, 1 2). The aim of this article is to prove algorithmically that there is no root s = σ + Iτ of Eta(s) with 0 < σ < 1 2. 1

This research article provides an unconditional proof of an inequality proposed by \textit{Srinivasa Ramanujan} involving the Prime Counting Function π(x),
(π(x))2<(ex/log x)π(x/e)
for every real x≥exp(547), using specific order estimates of the \textit{Mertens Function}, M(x). The proof primarily hinges upon investigating the underlying relation between M(x) and the \textit{Second Chebyshev Function}, ψ(x), in addition to applying the meromorphic properties of the \textit{Riemann Zeta Function}, ζ(s) with an intention of deriving an improved approximation for π(x).

We speculate upon an 'Unrecorded Proof' FUP of Fermat's Last Theorem FLT, which only assumes that every molecule of water can be treated as having an identical finite volume. We then show how the gedanken essentially mirrors, and can---albeit debatably----be treated as 'validating', Andrew Wiles' 'proof' of FLT as a pre-formal mathematical 'truth' in Markus Pantsar's sense.

This paper derives n-th Pythagorean relation from the edges of right triangle and the result be applied to other triangles as well as with the properties of binomial equations to discover the truly marvelous proof of Fermat's Last Theorem which the famous quotation French mathematician Pierre de Fermat quoted on the margin of his favorite book Diophantus' Arithmatica but the proof he never expressed. When the value of power n is equal to 2 FLT turns to Pythagorean Theorem, so the proof should be there [1]. If we can make a n-th power relation among the edges of right triangle, then by applying this to any triangle we will find our desire first step. For, nonhypotenuse integers [Appendix 6.1] general form of binomial equation is sufficient.

We study celestial amplitudes in string theory at one-loop. Celestial amplitudes describe scattering processes of boost eigenstates and relate to amplitudes in the more standard basis of momentum eigenstates through a Mellin transform. They are thus sensitive to both the ultraviolet and the infrared, which raises the question of how to appropriately take the field theory limit of string amplitudes in the celestial basis. We address this problem in the context of four-dimensional genus-one scattering processes of gluons in open string theory which reach the two-dimensional celestial sphere at null infinity. We show that the Mellin transform commutes with the adequate limit in the worldsheet moduli space and reproduces the celestial one-loop field theory amplitude expressed in the worldline formalism. The dependence on α′ continues to be a simple overall factor in one-loop celestial amplitudes albeit with a power that is shifted with respect to tree-level, thus making manifest that th...

The proof of Fermat's Last Theorem for exponent 3 is based on the fact that Fermat's Diophantine sum equation  x^3+y^3+z^3=0  is equivalent to the product form equation (3k)^3=(x+y+z)^3=3(x+y)(z+x)(z+y), where x+y,z+x,z+y are coprime integers.  The non existence of a common divisor means that two out of the three numbers  x+y,z+x,z+y  must be perfect cubes. 
The article was published in Czech language in the mathematical journal Kvaternion published by the Institute of Mathematics FSI VUT in Brno, ISSN 1805-1324 (printed version), ISSN 1805-1332 (on-line)
http://kvaternion.fme.vutbr.cz/2021/kv21_1-2_golan_web.pdf

The proof of Fermat's Last Theorem for exponent 3 is based on the fact that Fermat's Diophantine sum equation x 3 +y 3 +z 3 = 0 is equivalent to the product form equation (3k) 3 = (x+y+z) 3 = 3(x+y)(z+x)(z+y), where x+y, z+x, z+y are coprime integers. The non-existence of a common divisor means that two out of x+y, z + x, z + y must be perfect cubes. Without loss of generality, we can assume that 3 | z. Then it holds that x−3k = X 3 , y −3k = Y 3 , −z +3k = 9Z 3 and 3k = 3XY Z. This leads to the equations (−z) 3 = ((3Z) 3 (3Z 2 −XY)) 3 = 3(x+y)(x 2 −xy +y 2)/3 and (3Z 2 − XY) 3 = ((X 3 − Y 3)/2) 2 + 3((3Z 3)/2) 2. But every odd factor of the RHS must have the same form, so ((X 3 − Y 3)/2) 2 + 3((3Z 3)/2) 2 = (a 2 + 3b 2) 3 = (a 3 − 3ab 2 ± 6ab 2) 2 + 3(2a 2 b ∓ a 2 b ± 3b 3) 2 , gcd(a, b) = q, from where we get (Z/q) 3 = (2b/q)((a − b)/q)((a + b)/q) = A 3 B 3 C 3. This equation allows using of the proof method of infinite descent because A 3 + B 3 + C 3 = 0, where |ABC| < |y|.

This work contains two papers: the first published in 2022 and entitled "On the nature of some Euler's double equations equivalent to Fermat's last theorem" provides a marvellous proof through the so-called discordant forms of appropriate Euler's double equations, which could have entered in a not very narrow margin and the second instead published in 2024 and entitled "Some Diophantus-Fermat double equations equivalent to Frey's elliptic curve" provides the possible proof, which Fermat has not published in detail, but which uses the characteristic of all right-angled triangles with sides equal to whole numbers, or the famous Pythagorean identity. Some explanations in session(III) are provided: the first makes evident the nature of the "proof a' la Fermat" and the subsequent sessions clarify the direct and interesting connection of the two elementary proofs and it is necessary if you want to understand how two different elementary proofs of Fermat's Last Theorem are possible. Regarding the first paper, a method is used that simplifies Wiles' theory, a theory that has received much honors from the entire mathematical community. More precisely, through the aid of a Diophantine equation of second degree solved at first not directly, but as a consequence of the resolution of the double Euler equations that originated it and finally in a direct way, the author was able to obtain the following result: the intersection of the infinite solutions of Euler's double equations gives rise to an empty set and this only by exploiting a well-known Legendre Theorem, that is a criterion which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. It must be observed that this proof must in no way be interpreted as a sort of absurd revenge of elementary number theory over more modern analytic and algebraic treatments. The author himself has added a section in which he connects his concepts with some of those used by Wiles in his complex demonstration.

The focusing problem of lenses is examined and especially Descartes Ovals. A connection with Sahl's lens is examined. If the source of light is on the left of the sahle lens all rhe rays inside the lens are parallel. We examine a rotated Sahl lens and how to find the right side so thetotal optic length to be cosnstant

The article gives a new proof of Fermat's last theorem. The proof is based on the study of the properties of
natural numbers, an analysis of the constraints on the proposed solutions, and uses some general theorems on
the roots of algebraic equations. The connection between Fermat's theorem and Beal conjecture is discussed.
Beal conjecture is proved by induction using the same reasoning as in the proof of Fermat's theorem.

In this paper we will show you an elementary proof of Fermat’s Last Theorem, where ”the margin is too narrow to contain it”. We first start by showing you the cases n = 2, n = 3 and n = 4 to get you familiar with the general solution n ∈ Z, n ≥ 3. As a bonus, we will define a formula for each variable of the Pythagorean Triples (x, y, z) using the results from the case n = 2.

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In this paper, in Sections 1 and 2, we have described some equations and theorems concerning and linked to the Riemann zeta function. In the Section 3, we have showed the fundamental equation of the Riemann zeta function and the some equations concerning a new possible method for the calculation of the prime numbers. In conclusion, in the Section 4 we show the possible mathematical connections with various expressions of some sectors of String Theory and Number Theory and finally we suppose as the prime numbers can be identified as possible solutions to the some equations of the string theory (zeta string)

This beautiful piece of writing and perhaps one of my best discoveries, is intended to give an alternate but also a direct proof of the Fermat's last theorem. The theorem being one of the most popular, if not the most popular was also notorious for being one of the most difficult to prove. In 1994-about 357 years after it was first stated by Pierre de Fermat and accompanied with series of failed attempts from several mathematicians, came the supposed breakthrough, notably by Andrew Wiles¹. His approach required very sophisticated mathematical concepts in number theory, especially the modularity theorem². His approach however, isn't a actually a direct proof per se, it just happens to rely on the connection between distinct branches of mathematics (elliptic curves, modular forms and Fermat equation), just like Galois proof of the impossibility theorem relied on the connection between group theory and polynomial equations.

In this article, a different approach is utilized, somewhat unorthodox, as it requires new fundamental mathematical techniques which also happens to have implications for the philosophy and foundations of mathematics. The proof can be thought of as direct and it doesn't actually require much advanced mathematics as with the case of Wiles proof. The results established in this article could come in handy to proving other related theorems/conjectures and possibly advancing mathematics for the better.

The periodic Hurwitz zeta-function, a generalization of the classical Hurwitz zeta-function, is defined by a Dirichlet series with periodic coefficients and depends on a fixed parameter. We show that a wide class of analytic functions is approximated by shifts of a periodic zeta-function with rational parameter.

In this paper explicit analytical expression for Riemann Xi function ξ(s) is worked out for complex values of s. From this expression Riemann Hypothesis is proved. Analytic Expression for non-trivial Zeros of Riemann Zeta function ζ(s) is also found. A second proof of Riemann Hypothesis is also given.

Let E/Q be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let a E (p) = p + 1 − #E(F p) denote the trace of Frobenius. By the Hasse bound, a E (p) = 2 √ p cos θ p for a unique θ p ∈ [ 0, π ]. In this paper, we prove that the least prime p such that θ p ∈ [ α, β] ⊂ [ 0, π ] satisfies p N E β − α A , where N E is the conductor of E and the implied constant and exponent A > 2 are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime p ≡ a (mod q) for (a, q) = 1 satisfies p q L for an absolute constant L > 0.

English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time-such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the 'truly marvellous' proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the 'truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods which make use of Pythagoras theorem, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it. This proof is so elementary that anyone with a modicum of mathematical prowess in Fermat's days and in the intervening 358 years could have discovered this very proof. This brings us to the tentative conclusion that Fermat might very well have had the 'truly marvellous' proof which he claimed to have had and his 'truly marvellous' proof may very well have made use of elementary arithmetic methods.

The Fermat last theorem, defined in (2+1)-dimensional Minkowski spaces, is discussed and extended in natural and rational Mikowski's spaces. Several pieces of computational interest are given, with many practical examples. A definition of Fermat vector order, Fermat surfaces, and Fermat surface radius is given. Several conjectures are discussed, among them the existence of a Fermat theorem in (3+1)-dimensional Minkowski spaces.

Fermat´s Last Theorem can be regarded and proven as a mean value problem. Indeed, if the Fermat equation a^n + b^n  = c^n  is multiplied by 2 then c^n = (2a^n +2b^n )/2 is the arithmetic mean of  2a^n + 2b^n . In the following it will be shown that for n>2 the mean value c^n  never can be a perfect power of the same degree as a^n or b^n . From this immediately follows Fermat´s Last Theorem.

To analyze the solvability of Fermat's equation, the variables are replaced by interval values between them. By representing the equation in new variables, it is possible to establish the possible values of intervals. The solution of the 2 nd degree equation is found in new variables. The initial variables are represented only through intervals and a constant factor. The degree of the equation is reduced by dividing by one of the variables. The analysis of the obtained relations leads to the conclusion that the constant factor should be a unit value. Presenting the solution for higher powers similarly to the one for the 2 nd degree equation and using the equality of the constant factor 1, it is possible to show the absence of a solution for the powers of the equation greater than 2.

A divulgação da matemática é uma tarefa importante, diria mesmo, uma obrigação de todos quantos gostam de matemática e a ensinam a um nível superior. Mostrar, pelo menos a algum público, que a matemática não é aquele edifício monstruoso que lhe foi dado a conhecer quando frequentava os bancos da escola, apresentando-a vista de um prisma lúdico e desmontando, com textos simples, as barreiras que se colocam à sua compreensão, é um serviço louvável que se presta a esse público e à matemática. Esta parece ter sido a corajosa intenção dos autores. No entanto, a tarefa não foi totalmente conseguida. São vários os autores dos 43 pequenos textos, todos membros do Centro

In this paper we will show you an elementary proof of Fermat’s Last Theorem, where ”the margin is too narrow to contain it”. We first start by showing you the cases n = 2, n = 3 and n = 4 to get you familiar with the general solution n ∈ Z, n ≥ 3. As a bonus, we will define a formula for each variable of the Pythagorean Triples (x, y, z) using the results from the case n = 2.

The author of this paper claims many simple proofs of FLT are now accessible using recent discoveries involving Pascal's Triangle, and in particular, Moessner's Sieve.

Naive proofs of FLT abandon the constraint of the Natural number domain for Reals, unnecessarily converting a simple proof of 'that which is' into an impossible proof of 'that which is not'. The latter requires visibility into the entire solution set in the Real domain which is infinitely more vast than limiting the domain to only Naturals.

In this revised approach from 2015, the Pythagorean is used as the starting point to define a novel factorisation of  x^n  =  h^n - y^n  using a single factor (h-y) and additional instances of (h+y) for each increase of n.

This proof by contradiction clearly shows the how the inner terms of the expansion of  (h +y)^n  (h - y)  for all powers of n must all be zero to render FLT false, and why those conditions can only be met when n=2.

This is a Wiles independent proof using methods available to Fermat and is the first of two different simple approaches the author claims to be rigorous simple proofs of Fermat's Last Theorem.

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),…,ζ(s+ihrτ,F)) is proved. Here, h1,…,hr are algebraic numbers linearly independent over the field of rational numbers.

Unfortunately, however, the relation between a finite and an infinite is not always so straightfor-ward. The infinite and the finite mutually related as sheer others are inseparable. A related point is that while the infinite is determined in its own self by the other of itself, the finite, the finite itself is determined by its own infinite. Each of both is thus far the unity of its own other and itself. The inseparability of the infinite and the finite does not mean that a transition of the finite into the infinite and vice versa is not possible. In the finite, as this negation of the infinite, we have the sat-isfaction that determinateness, alteration, limitation et cetera are not vanished, are not sublated. The finite is a finite only in its relation to its own infinite, and the infinite is only infinite in its rela-tion to its own finite. As will become apparent, the infinite as the empty beyond the finite is bur-dened by the fact that determinateness, alteration, limitation et...

Answering the question of Jack Murt, in this paper, on the basis of specific examples from the history and practice of science, it is proved that often the completeness of the necessary evidence is determined by the level of dishonesty of the scientific community, which can be recognized as rigorously proven to be false, and absurdity to be true.

After understanding that the FLT can be presented onto the Cartesian Plane via a Symmetric Condition lead to a special construction, that leads to 2 method of proof: 1) A solving method via equations, can work for any n but of course can be proved solving the equations till the case n < 7, but it shows, intuitively, the reason why any case n > 2 produce the same known result: no integer triplets can be found. The aim of this work is to present in another following article another 2) Smarter method based onto my Complicate Modular Algebra (can be already seen onto the picture of the previous proof)

The &lt;strong&gt;division of zero by zero&lt;/strong&gt; turned out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.

Many different measures of association are used by medical literature, the relative risk is one of these measures. However, to judge whether results of studies are reliable, it is essential to use among other measures of association which are logically consistent. In this paper, we will present how to deal with one of the most commonly used measures of association, the relative risk. The conclusion is inescapable that the relative risk is logically inconsistent and should not be used any longer.

Objective. From a theoretical point of view, the demarcation between science and (fantastical) pseudoscience is in necessary for both practical and theoretical reasons. One specific nature of pseudoscience in relation to science and other categories of human reasoning is the resistance to the facts. Methods. Several methods are analyzed which may be of use to prevent that belief is masqueraded genuinely as scientific knowledge. Results. Modus ponens, modus tollens and modus conversus are reanalyzed. Modus sine, logically equivalent to modus ponens is developed. Modus inversus and modus juris are described in detail. Conclusions. In our striving for knowledge, there is still much more scientific work to be done on the demarcation between science and pseudoscience. Keywords: non strict inequality, quantum mechanics, science, pseudoscience

In view of many possible approaches to quantize the gravitational field already developed, it is possible to describe the dynamics of the gravitational field by the principles of quantum mechanics while following strictly the rules of Einstein’s theory of relativity. Thus far, quantum mechanics strictly predicts that all matter is quantum while general relativity describes the gravitational effects of classical matter. Then again, one might argue that we cannot reconcile general relativity with the principles of quantum mechanics at all. As we will see, it is possible to solve the problem of the quantization of the gravitational field in accordance with Einstein’s theory of realtivity.

Today, the division of zero by zero (0/0) is a concept in philosophy, mathematics and physics without a definite solution. On this view, we are left with an inadequate and unsatisfactory situation that we are not allowed to divide zero by zero while the need to divide zero by zero (i.e. divide a tensor component which is equal to zero by another tensor component which is equal to zero) is great. A solution of the philosophically, logically, mathematically and physically far reaching problem of the division of zero by zero (0/0) is still not in sight. The aim of this contribution is to solve the problem of the division of zero by zero (0/0) while relying on Einstein's theory of special relativity. In last consequence, Einstein's theory of special relativity demands the division of zero by zero. Due to Einstein's theory of special relativity, it is (0/0) = 1. As we will see, either we must accept the division of zero by zero as possible and defined, or we must abandon Einstein's theory of special relativity as refuted.

The principle of causality occupies an important place in the history of the philosophical interpretation of quantum mechanics from the beginning. In last consequence, today's Copenhagen dominated acausal interpretation of quantum mechanics casts doubt upon traditional views in the philosophy of nature and demands the revision of the principle of causality at a fundamental level of theoretical description in physics. Testing the mathematical and logical consistency of the quantum mechanical description of nature i. e. especially such as non-locality has become a subject of an ongoing dispute and research. Today, quantum-mechanical concepts i. e. such as non-locality refer to some mathematical foundations, especially to Bell's inequality and the CHSH inequality. Experimental data, analyzed by the help of Bell's inequality or the CHSH inequality favor a quantum mechanical description of nature, over local hidden variable theories (often referred to as local realism). In general, the use of mathematically inconsistent methods can imply a waste of money, time and effort on this account. Under some certain conditions (the assumption of independence) Bell's theorem and the CHSH inequality are already refuted. The purpose of this publication is explore the terra incognita, the interior logic that may lie beyond Bell's original theorem and the CHSH-inequality and to refute both, Bell's original theorem and the CHSH-inequality under any circumstances by the proof that we can derive a logical contradiction out of Bell's inequalities. Thus far, accept Bell's theorem or the CHSH-inequality as correct, then you must accept too that +0 = +1, which is a logical contradiction. Bell's theorem and the CHSH-inequality are refuted in general. In this insight, it appears to be necessary to revisit the very foundations of quantum theory and of physics as such.

The relationship between energy, time and space is still not solved in an appropriate manner. According to Newton's concept of time and space, both have to be taken as absolute. If we follow Leibniz and his arguments, space and time are relative. Since Einstein's theory of relativity we know at least that energy, time and space are deeply related. Albert Einstein originally predicted that time is nothing absolute but something relative, time changes and can change. Especially, time and gravitational field are related somehow even in detail if we still don't know how. According to the gravitational time dilation, the lower the gravitational potential, the more slowly time passes and vice versa. Somehow, it appears to be that the behaviour of time is directly linked to the behaviour of the gravitational field. The aim of this publication is to work out the interior logic between time and gravitational field and to make the proof that time is equivalent to the gravitational field and vice versa.

Adding another measure of relationship to the numerous already known measures of relationship makes only sense if the new one measure of relationship brings some advantages over the already known one. Methods. A new statistical method, the index of relationship (IOR), is developed which goes far beyond the odds ratio. As an example, the study of Loeb et al. (see also Loeb et al. 2017) has been re-analysed. Results. Evidence from different studies on the relationship between testosterone and prostate cancer has been inconsistent. Especially an exposure to testosterone and prostate cancer risk is very controversial. However, the data of the study of Loeb et al. (see also Loeb et al. 2017) support the null-hypothesis: without sexual activity no prostate cancer. Conclusion. Human prostate cancer is a sexually transmitted disease.

The article presents a simple proof of Fermat's Last Theorem (FLT) for a cube, obtained on the basis of the binomial expansion. The difference of two natural numbers having equal natural degrees certainly has a representation according to the incomplete binomial formula. It is proved that the cube of a natural number cannot be represented as an incomplete binomial, which means a simple proof of the FLT for = 3 is obtained.