Fermat's Theorem Research Articles

Sharing proofs with predicative theories through universe-polymorphic elaboration

As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism. It consists in trying to elaborate each term into a predicative universe-polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the elaboration, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in a partial algorithm which uses a constraint-postponement strategy for trying to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's library to Agda, including proofs of Bertrand's Postulate and Fermat's Little Theorem, which (as far as we know) were not available in Agda yet.

Asymptotic Fermat's last theorem for a family of equations of signature (2,2n,n)$(2, 2n, n)$

AbstractIn this paper, we study the integer solutions of a family of Fermat‐type equations of signature , . We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant such that if , there are no solutions of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.

108.20 Euler’s totient theorem and Fermat's little theorem are generalisations of one another!

108.20 Euler’s totient theorem and Fermat's little theorem are generalisations of one another!

Alessandro Bismuto.

"My favorite name theorem is Fermat's last theorem, probably one of the most complicated mathematical challenges of the last four centuries. I also love that he did not provide any demonstration as it would not have fit the margins … The most important quality of a mentor is adaptability-every student is different, and different is the way to them …" Find out more about Alessandro Bismuto in his Introducing… Profile.

Fermat's Theorem Arithmetic Demonstration

One of the factors of the number A+B-C is a fraction proper. Journal of Electrical Electronics Engineering ISSN: 2834-4928 40170 Mesos, France Theorem In a prime base n > 2 and reciprocals prime (coprime) numbers A, B, C, not multiples of n (basic case), the equality • An + Bn - Cn = 0 impossible.

Factoring primes and sums of two squares

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .

Великая теорема Ферма с точки зрения физика

The article is about the fact that the extraordinary beauty and conciseness of the formulation of Fermat's Last Theorem make us look for its visual solution. Let's try to consider Fermat's theorem from the eyes of physicist. Perhaps from this positions Pierre de Fermat found a solution whose main ideas would fit schematically in the fairly wide margins of the book, in a few drawings. Skeptics continue to believe that Pierre de Fermat was probably mistaken. Meanwhile, consistent application of the basic principles of physics, geometry, and engineering make us think differently.

Group Theory in Number Theory

The theory of groups exists in many fields of mathematics and has made a great impact on many fields of mathematics. In this article, this paper first introduces the history of group theory and elementary number theory, and then lists the definitions of group, ring, field and the most basic prime and integer and divisor in number theory that need to be used in this article. Then from the definitions, step by step, Euler's theorem, Bzout's lemma, Wilson's theorem and Fermat's Little theorem in elementary number theory are proved by means of definitions of group theory, cyclic groups, and even polynomials over domains. Finally, some concluding remarks are made. Many number theory theorems can be proved directly by the method of group theory without the action of tricks in number theory. Number theory is the thinking of certain special groups (e.g., (Z,+),(Z,)), so the methods of group theory work well inside number theory.

On the Geometrical Structure of Natural Numbers

This work studies the natural powers of prime numbers as the building blocks of a Euclidian vector semi space. Some vectors generate the composite natural numbers by defining an appropriate geometrical norm. One also studies the structure of extended Mersenne numbers within this geometric point of view. Further geometric applications and extensions of the powers of natural numbers are also studied with the help of inward vector operations. Two research lines follow the first discussion on the geometrical aspects of natural numbers: the extension of the Fermat theorem and the Euler-Riemann function.

A heuristic solution to Fermat's last theorem

A heuristic solution to Fermat's last theorem

Cryptography based on algebraic perpendicularities

Cryptography, the mathematics of protecting secret or sensitive information, is a continuously evolving area of interest. A concrete example of this is the fact that worldwide spending on information security and risk management technology and services has been estimated to reach over $150 billion in 2021. Modern cryptography is actually not a single separate domain of mathematics but an advanced encryption scheme can be based on applications of results in number theory, such as the Euler–Fermat Theorem, or involve the discrete logarithm problem using either a primitive root of a large prime or an element of an elliptic curve over a prime field. In the background, probability theory, statistics, studies on computational models and finite geometries, etc. play a major role. Recent research has considered even DNA-based molecular cryptography systems.

Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the infinitude of the primes

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

A constraints-based approach to Fermat's last theorem

A constraints-based approach to Fermat's last theorem

On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem

In this work, I provide a new rephrasing of Fermat’s Last Theorem, based on an earlier work by Euler on the ternary quadratic forms. Effectively, Fermat’s Last Theorem can be derived from an appropriate use of the concordant forms of Euler and from an equivalent ternary quadratic homogeneous Diophantine equation able to accommodate a solution of Fermat’s extraordinary equation. Following a similar and almost identical approach to that of A. Wiles, I tried to translate the link between Euler’s double equations (concordant/discordant forms) and Fermat’s Last Theorem into a possible reformulation of the Fermat Theorem. More precisely, through the aid of a Diophantine equation of second degree, homogeneous and ternary, solved not directly, but as a consequence of the resolution of the double Euler equations that originated it, I 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, which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. The impossibility of solving the second degree Diophantine equation thus obtained is possible using well-known techniques at the end of 18th century (see Euler, Lagrange and Legendre) and perhaps present in Fermat’s brilliant mind.

Special case of Fermat's Theorem

We prove under a mild condition that the only rationals x, y with x ≥ 0, y≥ 0 and x+y=N(k), for some k ∈ Q*, and xp+yp=1 are x=0, y=1 and x=1, y=0. Here, we let N denote the norm from Q(ωp) to Q for p an odd prime number.

An extension of Legendre's criterion in connection with the first case of Fermat's last theorem

An extension of Legendre's criterion in connection with the first case of Fermat's last theorem

A Novel Algorithmic Approach using Little Theorem of Fermat For Generating Primes and Poulet Numbers in Order

A Novel Algorithmic Approach using Little Theorem of Fermat For Generating Primes and Poulet Numbers in Order

Evaluation of quadratic residue with the help of Legendre symbol

Cryptography is the study of "Mathematical Systems," which includes two types of security protocols: privacy and authentication. Quadratic residue, a mathematical notion from the discipline of number theory known as Modular arithmetic, is extremely valuable in cryptography. Cryptography is deals with huge numbers, such as integers system with millions of digits or more. In this case, the Legendre symbols may be used to determine if an integer "x" has quadratic residue modulo "p" when p is Prime. This research article explains the mathematical ideas of quadratic residue, Fermat's little theorem, Euler's criteria, and the Legendre symbols. The main goal of this article is to explore the calculation problem of the number of solutions for one kind congruence equation modulo p (an odd prime) using simple methods and character sum properties, and to provide some interesting identities and asymptotic formulas for it.

Little Fermat Theorem Applying For Problems about Division

This article provides solutions to some divisibility problems using Fermat's little theorem. To have beautiful solutions for each of those problems, mathematicians have combined knowledge of: Theory of divisibility and division with remainder, greatest common divisor, least common multiple, prime numbers, congruences, exponentiation, etc. This helps students think positively and flexibly about their existing knowledge and skills, and present concise and creative solutions.

"Application of Fermat’s Little Theorem in Congruence Relation Modulo n"

According to Fermat's little theorem, for any p is a prime integer and 𝑔𝑐𝑑(𝑥, 𝑝) = 1, then the congruence 𝑥 𝑝−1 ≡ 1(𝑚𝑜𝑑 𝑛 )is true, if we remove the restriction that 𝑔𝑐𝑑(𝑥, 𝑝) = 1, we may declare𝑥 𝑝−1 ≡ 𝑥(𝑚𝑜𝑑 𝑝). For every integer x. Euler extended Fermat's Theorem as follows: if 𝑔𝑐𝑑(𝑥, 𝑝) = 1,then,where 𝑥 𝜙(𝑛) ≡ 1(𝑚𝑜𝑑 𝑛).𝜙 is Euler's phi-function. Euler's theorem cannot be implemented for any every integers x in the same manners as Fermat’s theorem works; that is, the congruence 𝑥 𝜙(𝑛)+1 ≡ 𝑥(𝑚𝑜𝑑 𝑛) is not always true. In this paper, we discussed the validation of congruence 𝑥 𝜙(𝑛)+1 ≡ 𝑥(𝑚𝑜𝑑 𝑛).