Fermat's Last theorem Algebraic Proof

In 1995, A. Wiles [1,2] announced a proof of Fermat's Last Theorem, which is stated as follows: If \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are relatively prime positive integers, then \(z^\pi\not=x^\pi+y^\pi.\) In this note, a simpler proof of this theorem is offered. It is proved that if \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are positive integers satisfying \(z^\pi=x^\pi+y^\pi,\) then \(x\), \(y,\) and \(z\) are each divisible by \(\pi\). The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.

Integers without large prime factors

Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connects these two equations by using the transformation equations. On solving the transformation equation we prove rst = 0, thus proving that only a trivial solution exists in the main equation r p + s p = t p .

On the periodic solutions emerging from the cylindrical precession of an axisymmetric satellite in a circular orbit

Structure analysis of periodically controlled chemical processes

This paper presents a k-ary Montgomery modular inverse algorithm over nonbinary computers by using Sedjelmaci's right shift k-ary greatest common divisor scheme. Over traditional binary computers, Kaliski's scheme can output Montgomery modular inverse Q − 12n mod P, where P is coprime to Q and n is the bit length of P. Over k-ary computers, our algorithm can discover the k-ary Montgomery inverse Q − 1km mod P, where P, Q, and k are pairwise relatively prime positive integers and m = log kP. In the worst case, the computational cost of our algorithm is O(m2)k-ary digit operations. Copyright © 2013 John Wiley & Sons, Ltd.

Let $P$ and $Q$ be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: $U_{0}(P,Q)=0,U_{1}(P,Q)=1,$ and $ U_{n+1}(P,Q)=PU_{n}(P,Q)+QU_{n-1}(P,Q)$ for $n\geq 1$ and $ V_{0}(P,Q)=2,V_{1}(P,Q)=P,$ and $V_{n+1}(P,Q)=PV_{n}(P,Q)+QV_{n-1}(P,Q)$ for $n\geq 1,$ respectively. In this paper, we assume that $P$ and $Q$ are relatively prime odd positive integers and $P^{2}+4Q>0.$ We determine all indices $n$ such that $U_{n}=(P^{2}+4Q)x^{2}.$ Moreover, we determine all indices $n$ such that $(P^{2}+4Q)U_{n}=x^{2}.$ As a result, we show that the equation $V_{n}^{2}(P,1)+V_{n+1}^{2}(P,1)=x^{2}$ has solution only for $n=2,$ $P=1,$ $x=5$ and $V_{n+1}^{2}(P,-1)=V_{n}^{2}(P,-1)+x^{2}$ has no solutions. Moreover, we solve some Diophantine equations.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Gorenstein simplices with a given [formula omitted]-polynomial

We consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z}$ in positive integers $x$, $y$ and $z$, where $a$, $b$ and $c$ are fixed pair-wise relatively prime positive integers greater than one. In this paper, we obtain several upper bounds for solutions $x$, $y$ and $z$ for which two of $x$, $y$ and $z$ are even. As their applications, we solve exponential Diophantine equations in which $a$, $b$ and $c$ are expressed as terms of linearly recurrence sequences.

In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one-to-one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x̄, p̄) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group 𝒞κ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group 𝒞κ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a κ component one. In this way we can map in a one-to-one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space, we get in an unambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies κ−1 and k−1. Furthermore, we also get the solutions of these equations, and the resulting representation is unitary. While our discussion is restricted to the problem mentioned above, in the concluding section we indicate our surmise for deriving systematically the unitary representation in quantum mechanics of arbitrary canonical transformations.

In this paper we present our views on algebra as a school subject.To say precisely what is the content of this subject, the language of abstract algebra has to be used.Accordingly, this content consists of the establishing and application of the properties of operations in the system of natural numbers which are then carried over to the extended systems.In the last of these extensions-the set of rational numbers, these properties can be taken to be axioms of ordered field and the main facts that are deduced from them.Of course, in the classroom practice, this content is didactically transformed and shaped to serve the learner and is aimed at the development of the skill of transforming numerical and literal expressions.On the other hand, the school arithmetic, when structured properly, consists of the gradual building of the number blocks up to 10, to 20, to 100 and to 1000, each of them having its own package of didactical tasks that are supposed to be accomplished.We take that both school arithmetic and school algebra have for their intuitive ground, the phenomenology which consists of sets at the sensory level and their configurations in the form of additive and multiplicative schemes.And while arithmetic deals with specific numbers denoted by their decimal notations, algebra deals with species of numbers (variables) denoted by letters.As for the properties of operations, in arithmetic they are related to specific schemes whose extent is determined by specific numbers, while in algebra they are related to the species of such schemes where letters replace specific numbers.The way we view school arithmetic and school algebra inspires us to suggest a fusion of these two subjects.In this paper we sketch how this fusion should be carried out, avoiding any possible abrupt semantic jumps from specific to general cases.The ultimate aim of teaching and learning both, arithmetic and algebra is the building of the systems of natural numbers, integers and rational numbers and as this aim is attained in the last classes of elementary school, in this paper we confine our considerations to the primary level.At this level we suggest and sketch: Elaboration of arithmetic in the form of gradual building

Blocks with a cyclic defect group

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

Cayley graphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> (d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k element subset A of Γ such that diam(Cay(Γ, A)) ≤ d. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam(Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> , A)) ≤ d. In this paper, we prove, among other results, that m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> (d, k) = m(d, k) for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley group is optimal with respect to its diameter and degree is always a cyclic group.