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.(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.

Conflicts of Interest

The authors declare no conflicts of interest.

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.

 [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.