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Integral Sequences of Infinite Length Whose Terms Are Relatively Prime

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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*)=1 for all positive integers

_{n}*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

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 24-28. doi: 10.4236/apm.2013.31005.

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