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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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.
|Additional Information:||© 1930 by the National Academy of Sciences. Communicated August 18, 1930. [H.T.E. was a] National Research Fellow.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Tony Diaz|
|Deposited On:||05 Apr 2008|
|Last Modified:||26 Dec 2012 09:55|
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