Special Dio3-Tuples for Pronic Number –I

Primes in tuples I

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance ap...

Our Special I. M. S. Number

At the time of the visit of Their Majesties KingGeorge and Qneen Mary to Calcutta in January 1912 the question was asked by one high up in authority what has the Indian Medical Service done for India ? At the time there was but little difficulty in giving a short answer to the question, and later Sir R. Havelock Charles, G.c.v.o. (not the least distinguished of the many I. M. S. names whom we h...

On the number of Diophantine m-tuples

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m = 2, 3 and 4. In this paper, we derive asymptotic extimates for the numb...

Chromatic number of graphs with special distance sets, I

Given a subset D of positive integers, an integer distance graph is a graphG(Z, D) with the set Z of integers as vertex set and with an edge joining two vertices u and v if and only if |u−v| ∈ D. In this paper we consider the problem of determining the chromatic number of certain integer distance graphs G(Z, D)whose distance set D is either 1) a set of (n + 1) positive integers for which the n ...

The Special Tuberculosis Number