Divisibility properties of certain recurrent sequences |
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Authors: | A Dubickas |
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Institution: | (1) Department of Mathematics and Informatics of Vilnius University, Institute of Mathematics and Informatics, Lithuania |
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Abstract: | Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence
x0 = g, xn = x
n−1
n
+ F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number,
is considered. We show, for instance, that the sequence x0 = 2, xn = x
n−1
n
+ 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except
for three prime numbers p = 23, 47, 167 that do not divide xn. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences ζn!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 76–82. |
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Keywords: | |
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