**Author(s):**Kazuyuki Hatada

**Journal:**Advances in Pure Mathematics

ISSN 2160-0368

**Volume:**03;

**Issue:**01;

**Start page:**24;

**Date:**2013;

Original page

**Keywords:**Relatively Prime | Integral Sequences of Infinite Length | Sets of Infinitely Many Prime Numbers

**ABSTRACT**

It is given in Weil and Rosenlicht ([1], p. 15) that (resp. 2) for all non-negative integers

*m*and

*n*with

*m≠n*if

*c*is any even (resp. odd) integer. In the present paper we generalize this. Our purpose is to give other integral sequences such that G.C.D.(

*ym*,

*yn*)=1 for all positive integers

*m*and

*n*with

*m≠n*. Roughly speaking we show the following 1) and 2). 1) There are infinitely many polynomial sequences such that G.C.D.(

*f*

*m*(

*a*),

*f*

*n*(

*a*))=1 for all positive integers

*m*and

*n*with with

*m≠n*and infinitely many rational integers

*a.*2) There are polynomial sequences such that G.C.D.(

*g*

*m*(

*a,b*),

*g*

*n*(

*a,b*))=1 for all positive integers

*m*and

*n*with

*m≠n*and arbitrary (rational or odd) integers

*a*and

*b*with G.C.D.(

*a*,

*b*)=1. Main results of the present paper are Theorems 1 and 2, and Corollaries 3, 4 and 5.