The Journal of the Indian Mathematical Society

S. Sivasankaranarayana Pillai ¹

Affiliations
1 Annamalai University, IN

Abstract

The object of this paper is to prove the following theorems:-

Theorem I. If m, n, a, b, c are given positive integers, given a positive number δ, we can find a number x (δ), such that

am^z - bn^r > m^x(1-δ)

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S. Sivasankaranarayana Pillai

Abstract

Let S = ∑ _nφ (nv + c), [nv + c ^ a>]

where

(x) is the number of integers less than and prime to x. The object of this note is to prove the following Theorm: S = ³/_π2, φ (v).x² + 0(z log x).

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S. Sivasankaranarayana Pillai ¹

Affiliations
1 University of Madras, IN

Abstract

In a thesis submitted to the California University, Henry W. Stager discusses the properties of numbers which contain no factors of the form p (kp + 1), where p is a prime and ^ is a positive integer. He conjectures that the number of sucb numbers, (P's as he calls them) and the number of primes within a given limit differ asymptotically by a constant.

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S. Sivasankaranarayana Pillai ¹

Affiliations
1 Annamalai University, IN

Abstract

Let G¹ (k) be the minimum number of positive kth powers of primes required to represent every number from a certain point onwards. It is very difficult to prove the existence of G¹ (k). We require here a combination of weapons used in proving Goldbach's theorem and Waring's problem.

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S. Sivasankaranarayana Pillai ¹

Affiliations
1 Annamalai University, IN

Abstract

We make use of the following two well-known lemmas :-

LEMMA (1 1). if (x, y) = 1, then x ± y and

 x^2r± x^2r-1y+x^2r-2 y²  +x^2r-3 y³+ …..y ^2r

cannot have any common factor except the di vision of r + 1.

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