Diophantus' 20th Problem and Fermat's Last Theorem for n=4: Formalization of Fermat's Proofs in the Coq Proof Assistant

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.Comment: 16 page

Review of Fermat’s Last Theorem

A review of Fermat’s last theorem form Fermat to Wiles is presented. A review of Beal’s conjecture is also presented

History of Fermat\u27s Last Theorem

Around 1637, Pierre de Fermat made a now-famous mathematical conjecture. However, Fermat\u27s conjecture neither began nor ended with him. Fermat\u27s last theorem, as the conjecture is called, has roots approximately 3600 years old. The proof of the theorem was not realized until 1994, over 350 years after it was proposed by Fermat

Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980. What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888

A study of Fermat\u27s last theorem.

This work is intended primarily for the student of mathematics who may not necessarily wish to delve into the complete theory of the problem. but who may wish to get a picture of the work necessary to study and understand it. For a complete history of the problem until 1919, refer to Chapter XXVI of Dickson\u27s History of the Theory of Numbers, II, and for the work primarily connected with Irregular Cyclotomic Fields and Fermat\u27s Last Theorem through part of 1927, refer to Algebraic Numbers, II, National Research Council Bulletin, # 62. For a Short presentation of the problem refer to Mordell, Three Lectures on Fermat\u27s Last Theorem. For a more detailed analysis of the methods used to work on this problem refer to Bachmann, Das Fermatproblem. The author of this treatise has attempted to cover certain points. First, to give the nature of Fermat\u27s Last Theorem. Second, an introduction. which will contain some historical facts of the Last Theorem, a reference to the mathematical development which resulted from this problem and its status at the present. Third, a detailed account of the method of infinite descent. The cases n = 3, 4. 5 are used to show this method of attack. For n = 5 the procedure is that of Dirichlet. In order that a better understanding may be secured of various steps in his proof. all the theorems of his article are reproduced without proof. Fourth. an introduction to the theory of ideals. This is not an exhaustive study of ideals but an attempt to aid the beginner to understand them

Proving The Fermat Last Theorem for Case q≤n

Fermat's Last Theorem is a well-known classical theorem in mathematics. Andrew Willes has proven this theorem using the modular elliptic curve. However, the proposed proof is difficult for mathematicians and researchers to understand. For this reason, in this study, we provide evidence of several properties of Fermat's Last Theorem with a simple concept. We use Newton's Binomial Theorem, well-known in Fermat's time. In this study, we prove Fermat's Last Theorem for case . We also use the Newton’s Binomial theorem to verify several cases .