Pythagorean triples containing generalized Lucas numbers

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.

In 1995, A, Wiles announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat's Last Theorem, which is stated as follows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, an elegant proof of this result is given. It is proved, using elementary algebra, that if is an odd prime and x; y; z are positive integers satisfying z = x+y; then z; x; y are each divisible by :

Integers without large prime factors

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes 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.

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer; z and z2 irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations we prove the theorem.

Abstract.Let a1,;… a9 be nonzero integers and n any integer. Suppose that a1+…+a9 ≡ n (mod 2) and (ai ; aj ) = 1 for 1 ≤ i < j ≤9. In this paper we prove the following:(i) If aj are not all of the same sign, then the cubic equation has prime solutions satisfying pj ≪|n|1/3 + max{|aj|}14+∊.(ii) If all aj are positive and n ≫ max{|aj|} 43+∊, then is solvable in primes pj.These results are an extension of the linear and quadratic relative problems.

Let G be a simple graph of order n. A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in G are co-neighbour vertices if they share the same neighbours. Clearly, if S is a set of pairwise co-neighbour vertices of a graph G, then S is an independent set of G. Let c=a+b√m and c=a−b√m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. In [M. Lepovic, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730-738, 2007], the author defined the matrix Ac(G)=[cij]n to be the conjugate adjacency matrix of G, if cij=c for any two adjacent vertices i and j, cij=c for any two nonadjacent vertices i and j,and cij= 0 if i=j. In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices.

Two well-studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression or in geometric progression. We study generalizations of this problem by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence \(x, f_n(x), f_n(f_n(x))\), where \(f_n(x) = x + n\), n a nonzero integer. Thus, avoiding 3-term arithmetic progressions are equivalent to containing no three elements of the form \(x, f_n(x), f_n(f_n(x))\) with \(f_n \in \mathscr {F}_\mathrm{t}\), the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions (\(f_n(x) = nx\) with \(n > 1\) a natural number) and exponential progressions (\(f_n(x) = x^n\)). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1 and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density; however, we consider several different notions of density, including the uniform and exponential densities.

Gorenstein simplices with a given [formula omitted]-polynomial

Let [Formula: see text] be an integer, [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] are nonzero integers, and let [Formula: see text] be an integer. Suppose that [Formula: see text] satisfy some necessary congruent conditions. In this paper, it is proved that (i) if [Formula: see text] are not all of the same sign, then the equation [Formula: see text] has prime solutions satisfying [Formula: see text], (ii) if all [Formula: see text] are positive and [Formula: see text], then [Formula: see text] is solvable in primes [Formula: see text], where [Formula: see text]. Our result uses [Formula: see text] as an improvement of the recent result [Formula: see text] due to Zhao (2016), and largely improves the results [Formula: see text] for [Formula: see text] proved by Yang and Hu (2016).

For any non-negative integer n and non-zero integer r, let $$p_r(n)$$ denote Ramanujan’s general partition function. In this paper, we prove many infinite families of congruences modulo 7 for the general partition function $$p_r(n)$$ for negative values of r by using q-identities.