On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q) is defined by U_0=0, U_1=1, U_n= P*U_{n-1}-Q*U_{n-2} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,...,7, U_n is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,...,12, the only non-degenerate sequences where gcd(P,Q)=1 and U_n(P,Q)=square, are given by U_8(1,-4)=21^2, U_8(4,-17)=620^2, and U_12(1,-1)=12^2.Comment: 11 pages. To appear in Journal of Number Theor

Computing all integer solutions of a genus 1 equation

The Elliptic Logarithm Method has been applied with great successto the problem of computing all integer solutions of equations ofdegree 3 and 4 defining elliptic curves. We extend this methodto include any equation f(u,v)=0 that defines a curve of genus 1.Here f is a polynomial with integer coefficients and irreducible overthe algebraic closure of the rationals, but is otherwise of arbitrary shape and degree.We give a detailed description of the general features of our approach,and conclude with two rather unusual examples corresponding to equationsof degree 5 and degree 9.Elliptic curve;Elliptic logarithm;Dophantine equation

Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bibliography has been considerably enriched. This article is of an expository nature.Comment: A number of misprints have been corrected and one or two improvements have been done to the previous version of the paper with same title. The paper will appear to Elem. der Mat

On px2+q2n=yppx^2 + q^{2n}= y^p and related Diophantine equations

The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation turns out to be a difficult task. We focus our attention to $p=5$, by reason of an article by F. Abu Muriefah, published in this journal, vol. 128 (2008), 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equation $5x^2-4=y^n$ (integer solutions), interesting in themselves, which are exposed in Sections 3 and 4. These last results are obtained by using tools such as Linear Forms in Two Logarithms and Hypergeometric Series.Comment: 23 pages, second version with minor revision

Computing all integer solutions of a genus 1 equation

The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u,v)=0 that defines a curve of genus 1. Here f is a polynomial with integer coefficients and irreducible over the algebraic closure of the rationals, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree 5 and degree 9

Cavitation erosion damage of scroll steel plates by high-speed gas working fluid

A steel plate is one of the critical components of a scroll expander system that can experience cavitation micro-pitting while in service. The content of the present paper consists of two distinct but interrelated parts. The first part aims to highlight that the use of Computational Fluid Dynamics (CFD) simulations in conjunction with experimental measurements can constitute a quite promising tool for the prediction of cavitation erosion areas in scroll expander systems. For this purpose a three-dimensional CFD, steady state numerical simulation of the refrigerant working fluid is employed. Numerical results revealed the critical areas where cavitation bubbles are formed. These numerical critical areas are in direct qualitative agreement with the actual eroded regions by cavitation, which were found by microscopic observations across the steel plate on an after use, scroll expander system. The second part of the paper, aims to further investigate the behaviour and the durability of the steel plate of the studied scroll expander system subjected to cavitation erosion by using an ultrasonic experimental test rig. Scanning Electron Microscopy (SEM) and optical interferometer micrographs of the damaged surfaces were observed, showing the nature of the cavitation erosion mechanism and the morphological alterations of the steel plate samples. Experimental results are explained in terms of the cavitation erosion rates, roughness profile, accumulated strain energy, and hardness of the matrix. The experimental study can serve as a valuable input for future development of a CFD numerical model that predicts both cavitation bubbles formation as well as cavitation damage induced by the bubbles that implode on the steels plates

On Sums of Consecutive Squares

In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in this sum is the square of k. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstrass equation with parameter k. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms of k. We conjecture that this point indeed generates the free part of the Mordell-Weil group, and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. For k in the range 1...100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameter k. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms - which is an essential ingredient of our approach - the rank is the dominant factor. Also the extra computational effort that is needed for some values of k in order to determine the rank unconditionally and construct a set of generators for the Mordell-Weil group deserves special attention, as there are some unusual features