MATHEMATICA BOHEMICA, Vol. 132, No. 2, pp. 137-175 (2007)

Bounds for frequencies of residues of
second-order recurrences modulo $p^r$

Walter Carlip, Lawrence Somer

Walter Carlip, Department of Mathematics, Franklin & Marshall College, Lancaster, Pennsylvania  17604, USA; mailing address: 408 Harvard Street, Vestal, New York  13850, USA, e-mail:; Lawrence Somer, Department of Mathematics, Catholic University of America, Washington D. C. 20064, USA, e-mail:

Abstract: The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors' results have application to the $p$-stability of second-order recurrence sequences.

Keywords: Lucas, Fibonacci, stability, uniform distribution, recurrence

Classification (MSC2000): 11B37, 11A25, 11A51, 11B39

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