Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.Comment: 9 pages; Keywords: Elliptic curves over finite fields, rational point

A p-adic look at the Diophantine equation x^{2}+11^{2k}=y^{n}

We find all solutions of Diophantine equation x^{2}+11^{2k} = y^{n} where x>=1, y>=1, n>=3 and k is natural number. We give p-adic interpretation of this equation.Comment: 4 page

On the spectral radius of bipartite graphs which are nearly complete

For p, q, r, s, t is an element of Z(+) with rt <= p and st <= q, let G = G(p, q; r, s; t) be the bipartite graph with partite sets U = {u(1), ..., u(p)} and V = {v(1),..., v(q)} such that any two edges u(i) and v(j) are not adjacent if and only if there exists a positive integer k with 1 <= k <= t such that (k - 1) r + 1 <= i <= kr and (k - 1) s + 1 <= j <= ks. Under these circumstances, Chen et al. (Linear Algebra Appl. 432: 606-614, 2010) presented the following conjecture: Assume that p <= q, k < p, vertical bar U vertical bar = p, vertical bar V vertical bar = q and vertical bar E(G)vertical bar = pq - k. Then whether it is true that lambda(1)(G) <= lambda(1)(G(p, q; k, 1; 1)) = root pq - k + root p(2)q(2) - 6pqk + 4pk + 4qk(2) - 3k(2)/2. In this paper, we prove this conjecture for the range min(vh is an element of V){deg v(h)} <= left perpendicular p-1/2right perpendicular.BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of KoreaMinistry of Education & Human Resources Development (MOEHRD), Republic of Korea; Research Project Offices of UludagUludag University; Selcuk UniversitiesSelcuk UniversityThe first author is supported by BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea, and the other authors are partially supported by Research Project Offices of Uludag (2012-15 and 2012-19) and Selcuk Universities

Preface

On behalf of the International Advisory Board and the Local Organizing Committee of The International Congress in Honour of Professor Hari M. Srivastava, which was held in the Auditorium at the Campus of Uludag University, Bursa, Turkey on August 23-26, 2012, and on our own behalves, we would like to express our happiness and gratitude to all participants who attended and actively participated in our Congress. This article is being published in each of the four Special Issues of the SpringerOpen journals, Advances in Difference Equations, Boundary Value Problems, Fixed Point Theory and Applications and Journal of Inequalities and Applications, which are entitled Proceedings of the International Congress in Honour of Professor Hari M. Srivastava