APM  Vol.3 No.1 , January 2013
Integral Sequences of Infinite Length Whose Terms Are Relatively Prime
Author(s) Kazuyuki Hatada*
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.(fm(a),fn(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.(gm(a,b),gn(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.


Cite this paper
K. Hatada, "Integral Sequences of Infinite Length Whose Terms Are Relatively Prime," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 24-28. doi: 10.4236/apm.2013.31005.
References
[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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