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Shorey’s influence in the theory of irreducible polynomials
 in: N. Saradha (Ed.), Diophantine Equations, Narosa Publ
, 2008
"... Dedicated to Tarlok N. Shorey and his continuing contributions to Number Theory ..."
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Dedicated to Tarlok N. Shorey and his continuing contributions to Number Theory
A Generalization Of An Irreducibility Theorem Of I. Schur
, 1991
"... this paper refers to irreducibility over the rationals. Some condition, such as ja 0 j = ja n j = 1, on the integers a j is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial in Z[x] is irreducible (which is clearly not the case). In this paper ..."
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this paper refers to irreducibility over the rationals. Some condition, such as ja 0 j = ja n j = 1, on the integers a j is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial in Z[x] is irreducible (which is clearly not the case). In this paper, we will mainly be interested in relaxing the condition ja n j = 1. Specifically, we will show:
Generalizations of some irreducibility results by Schur
, 2009
"... Let a ≥ 0 and a0, a1,...,an be integers with ..."
PRIME FACTORS OF ARITHMETIC PROGRESSIONS AND BINOMIAL COEFFICIENTS
"... Sylvester [Syl] proved in 1892 that a product of k consecutive positive integers x, x + 1,..., x + k − 1 greater than k is divisible by a prime exceeding k. It is a generalization of Bertrand’s Postulate that there is a prime among k+1, k+2,..., 2k. (Take x = k + 1.) The assumption x> k can not be r ..."
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Sylvester [Syl] proved in 1892 that a product of k consecutive positive integers x, x + 1,..., x + k − 1 greater than k is divisible by a prime exceeding k. It is a generalization of Bertrand’s Postulate that there is a prime among k+1, k+2,..., 2k. (Take x = k + 1.) The assumption x> k can not be removed since x = 1 should be
On Ramachandra’s Contributions to Transcendental Number Theory
, 2009
"... The title of this lecture refers to Ramachandra’s paper in Acta Arithmetica [36], which will be our central subject: in section 1 we state his Main Theorem, in section 2 we apply it to algebraically additive functions. Next we give new consequences of Ramachandra’s results to density problems; for i ..."
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The title of this lecture refers to Ramachandra’s paper in Acta Arithmetica [36], which will be our central subject: in section 1 we state his Main Theorem, in section 2 we apply it to algebraically additive functions. Next we give new consequences of Ramachandra’s results to density problems; for instance we discuss the following question: let E be an elliptic curve which is defined over the field of algebraic numbers, and let Γ be a finitely generated subgroup of algebraic points on E; is Γ dense in E(C) for the complex topology? The other contributions of Ramachandra to transcendental number theory are dealt with more concisely in section 4. Finally we propose a few open problems. The author wishes to convey his best thanks to the organizer of the Madras Conference of July 1993 in honor of Professor Ramachandra’s 60th birthday, R. Balasubramanian, for his invitation to participate, which provided him the opportunity to write this paper. Next he is grateful to the organizer of the Bangalore Conference of December 2003 in honor of Professor Ramachandra’s 70th birthday, K. Srinivas, for his invitation to participate, which provided him the opportunity to publish this