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 nn−1 + 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 nn−1 + 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.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 76–82.
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Dubickas, A. Divisibility properties of certain recurrent sequences. J Math Sci 137, 4654–4657 (2006). https://doi.org/10.1007/s10958-006-0261-0
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DOI: https://doi.org/10.1007/s10958-006-0261-0