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Periodicity in sequences defined by linear recurrence relations

Engstrom, H. T. (1930) Periodicity in sequences defined by linear recurrence relations. Proceedings of the National Academy of Sciences of the United States of America, 16 (10). pp. 663-665. ISSN 0027-8424. http://resolver.caltech.edu/CaltechAUTHORS:ENGpnas30

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Abstract

A sequence of rational integers u0, u1, u2, ...(1) is defined in terms of an initial set u0, u1, ..., uk-1 by the recurrence relation un+k + a1un+k-1 + ... + akun = a, n ≥ 0, (2) where a1, a2, ..., ak are given rational integers. The author examines (1) for periodicity with respect to a rational integral modulus m. Carmichael (1) has shown that (1) is periodic for (ak, p) = 1 and has given periods (mod m) for the case where the prime divisors of m are greater than k. The present note gives a period for (1) (mod m) without restriction on m. The results include those of Carmichael. The author also shows that if p divides ak (1) is periodic after a determined number of initial terms and obtains a period.


Item Type:Article
Additional Information:© 1930 by the National Academy of Sciences. Communicated August 18, 1930. [H.T.E. was a] National Research Fellow.
Record Number:CaltechAUTHORS:ENGpnas30
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:ENGpnas30
Alternative URL:http://www.pnas.org/cgi/reprint/16/10/663
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10004
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:05 Apr 2008
Last Modified:26 Dec 2012 09:55

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