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Integral Sequences of Infinite Length Whose Terms Are Relatively Prime ()

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

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

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | A. Weil and M. Rosenlicht, “Number Theory for Beginners,” Springer Verlag, New York, 1979. doi:10.1007/978-1-4612-9957-8 |

[2] | G. H. Hardy and E. M. Wright, “An Introduction to the Theory of Numbers,” 4th Edition, Oxford University Press, Ely House, London, 1971. |

[3] | A. Baker, “A Concise Introduction to the Theory of Numbers,” Cambridge University Press, Cambridge, 1984. doi:10.1017/CBO9781139171601 |

[4] | B. J. Birch, “Cyclotomic Fields and Kummer Extensions,” In: J. W. S. Cassels and A. Fr?hlich, Eds., Algebraic Number Theory, Academic Press, London, 1967, pp. 85-93. |

[5] | S. Lang, “Algebraic Number Theory,” Addison-Wesley Publishing Company, Massachusetts, 1970. |

[6] | S. Lang, “Algebra,” 3rd Edition, Springer Verlag, New York, 2002. doi:10.1007/978-1-4613-0041-0 |

[7] | E. Weiss, “Algebraic Number Theory,” 2nd Edition, Chelsea Publishing Company, New York, 1976. |

[8] | H. Weyl, “Algebraic Theory of Numbers,” Princeton University Press, Princeton, 1940. |

[9] | J.-P. Serre, “Local Fields,” Springer Verlag, New York, 1979. |

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