Integral Sequences of Infinite Length Whose Terms Are Relatively Prime

Kazuyuki Hatada

Department of Mathematics, Faculty of Education, Gifu University, Gifu, Japan.

DOI: 10.4236/apm.2013.31005                

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.

Keywords

Relatively Prime; Integral Sequences of Infinite Length; Sets of Infinitely Many Prime Numbers

Conflicts of Interest

The authors declare no conflicts of interest.

References

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