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In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127), an elementary and thoroughly arithmetical proof of Fermat's last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, secon d part of the paper is outlined, getting directed to an eventual proof of the case “n = 3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it.

In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127), an elementary and thoroughly arithmetical proof of Fermat's last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, secon d part of the paper is outlined, getting directed to an eventual proof of the case “n = 3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it.

One shows that the Last Fermat Theorem is equivalent to the statement that all rational solutions of the famous equation are provided by an orbit of rationally parametrized subgroup of a group preserving k-ubic form. This very group naturally arrises in the generalized Clifford algebras setting .

Includes bibliographical references (p. 139-140) and index

Includes bibliographical references (p. 139-140) and index

Includes bibliographical references (p. 139-140) and index

"Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions. [Abstract]" is an article from Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London, Volume 2 . View more articles from Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-109880

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

"Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers" is an article from The Annals of Mathematics, Volume 18 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007234

"Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers" is an article from The Annals of Mathematics, Volume 18 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007234

In the present paper we study, in a mathematically non-formal way, the validity of the Fermat's Last Theorem (FLT) by generalizing the usual procedure of extracting the square root of non convenient objects initially introduced by P. A. M. Dirac in the study of the linear relativistic wave equation.

We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effective. Then we propose some modest ways of study in a more diophantine context using radicals; this point of view would require further nonalgebraic investigations.

Speaker : Robert Osserman, Lenore Blum, Ken Ribet, John Conway, Lee Dembart

Speaker : Robert Osserman, Lenore Blum, Ken Ribet, John Conway, Lee Dembart

"Errata: On the Class Number of the Field Ω (e 2iπ /p n ) and the Second Case of Fermat's Last Theorem" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 6 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-84224

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189.

Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions. Knight, T Philosophical Transactions of the Royal Society of London (1776-1886). 1817-01-01. 107:245–251

`Fermat's Last Theorem for the exponent 3 has received numerous proofs, the most common of which being either in Euler's or in Gauss' style. This latter works entirely in the ring of integers of the quadratic field generated by the square root of -3. A proof in Euler's style is based on a central lemma related to properties of the quadratic form x^2 + 3y^2, then it proceeds by descent by considering Two main cases. In the present version, the central lemma receives a short proof and the classical proof by descent in Two main cases boils down to a One-case proof.

We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the "wonderful" method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller non-negative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are bounded by zero. We describe the formalization of this proof which has been carried out in the Coq proof assistant. Moreover, as a direct and no less historical application, we also provide the proof (by Fermat) of Fermat's last theorem for n=4, as well as the corresponding formalization made in Coq.

Despite of an active work of many researchers in the theory of quantum computations, this area still saves some mysterious charm. It is already an almost common idea, that maybe many fashionable current projects will fade in future, but some absolutely unpredictable applications appear instead. Why such optimistic predictions are legal here, despite of an extreme difficulty to suggest each one new promising quantum algorithm or realistic "industrial" application? One reason -- is very deep contents of this area. It maybe only an extremely unlucky occasion, if such a fundamental thing won't supply us with some bright insights and serious new applications. A sign of such nontrivial contents of a theory -- are unexpected links between different branches of our knowledge. In the present paper is mentioned one such link -- between application of Weyl quantization in the theory of quantum computations and abstract mathematical constructions born in mid of XIX century due to unsuccessful tries to prove Fermat's last theorem.

Includes bibliographical references (p. 77-78)

Includes bibliographical references (p. 77-78)

Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.

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.

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.

scan of book Fermat’s Enigma: The quest to prove Fermat's last theorem by Albert Violant i Holz [2012] {512.74--IA} Everything is mathematical Issue 9

Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions. Knight, T Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London (1800-1843). 1815-01-01. 2:69–70

Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions. Knight, T Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London (1800-1843). 1815-01-01. 2:69–70

Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions. Knight, T Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London (1800-1843). 1815-01-01. 2:69–70

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

"Note Respecting the Demonstration of the Binomial Theorem Inserted in the Last Volume of the Philosophical Transactions" is an article from Philosophical Transactions of the Royal Society of London, Volume 107 . View more articles from Philosophical Transactions of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-107584

"On the Class Number of the Field Ω (e 2iπ /p n ) and the Second Case of Fermat's Last Theorem" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 6 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-84157

"On the Class Number of the Field Ω (e 2iπ /p n ) and the Second Case of Fermat's Last Theorem" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 6 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-84157

Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem. This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)! QUESTIONS: 1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it? 2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory? 3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it? 4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)? 5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this? 6. Does anyone understand Andrew Wiles’ paper? LOL SOURCES I USED TO STUDY: Keith Conrad’s Lectures on Modular Forms (8 part video series): https://www.youtube.com/watch?v=LolxzYwN1TQ Keith Conrad’s Notes on Modular Forms: https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf “Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo. (The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.) “The Arithmetic of Elliptic Curves” by Joseph Silverman HOMEWORK IDEA CREDIT goes to Looking Glass Universe! SAGE RESOURCES: “Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html Download SAGE: https://www.sagemath.org/download.html Proof of the Hasse-Weil Bound on Terry Tao’s Blog: https://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/ OTHER VIDEOS ON THESE TOPICS: Numberphile Playlist: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlW Elliptic Curves and Modular Forms: https://www.youtube.com/watch?v=A8fsU97g3tg SOFTWARE USED TO MAKE THIS VIDEO: SAGE for the code and the graphs https://github.com/hernanat/dcolor for domain coloring Adobe Premiere Elements For Video Editing MUSIC: Music Info: Documentary - AShamaluevMusic. Music Link: https://www.ashamaluevmusic.com Follow me! Twitter: https://twitter.com/00aleph00 Instagram: https://www.instagram.com/00aleph00 Intro: (0:00) Elliptic Curves: (0:58) Modular Forms: (3:26) Taniyama Shimura Conjecture: (7:26) Fermat's Last Theorem: (8:02) Questions for you!: (8:51)