Integral Sequences of Infinite Length Whose Terms Are Relatively Prime
- 1 Department of Mathematics, Faculty of Education, Gifu University, Gifu, Japan
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.( y m , y n )=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.
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