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. https://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 | ||||||
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Additional Information: | © 1930 by the National Academy of Sciences. Communicated August 18, 1930. [H.T.E. was a] National Research Fellow. | ||||||
Issue or Number: | 10 | ||||||
Record Number: | CaltechAUTHORS:ENGpnas30 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:ENGpnas30 | ||||||
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: | 03 Oct 2019 00:05 |
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