Representing Primes as the Form x2+ny2x^2+ny^2 in Some Imaginary Quadratic Fields

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.Comment: 8 pages, This paper has been withdrawn by the author since it was merged into the article arXiv:1405.5776 on August 8, 201

SMARANDACHE FUNCTION JOURNAL, 1

This journal is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies. SNJ is a referred journal: reviewed, indexed, cited, concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc

On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture holds for the equations x2−xy+ax+by+cz=0 and x2−y2+ax+by+cz=0 for a,b,c∈Z such that abc=0 or a+b+c=0. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in Z such as x,x+y,xy+x+y, building on the recent result on the partition regularity of x,x+y,x

Whiskered parabolic tori in the planar (n+ 1) -body problem

“This is a post-peer-review, pre-copyedit version of an article published in Communications in mathematical physics. The final authenticated version is available online at: https://dx.doi.org/10.1007/s00220-019-03507-3The planar (n+1)-body problem models the motion of n+1 bodies in the plane under their mutual Newtonian gravitational attraction forces. When n=3, the question about final motions, that is, what are the possible limit motions in the planar (n+1)-body problem when t¿8, ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar (n+1)-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocity while the rest perform a bounded motion. These solutions are related to whiskered parabolic tori at infinity, that is, parabolic tori with stable and unstable invariant manifolds which lie at infinity. These parabolic tori appear in cylinders which can be considered “normally parabolic”. The existence of these whiskered parabolic tori is a consequence of a general theorem on parabolic tori developed in this paper. Another application of our theorem is a conjugation result for a class of skew product maps with a parabolic torus with its normal form generalizing results of Takens (Ann Inst Fourier 23(2):163–195, 1973), and Voronin (Funktsional Anal i Prilozhen 15(1):1–17, 96, 1981).Peer ReviewedPostprint (author's final draft

Elliptic curve over totally real fields: A Survey

In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along with certain applications.Comment: This is a survey for Bhaskaracharya Pratishthana (BP) program proceedings. Any comments are welcom