Multiplicities of second order linear recurrences
Authors:
Ronald Alter and K. K. Kubota
Journal:
Trans. Amer. Math. Soc. 178 (1973), 271284
MSC:
Primary 10A35; Secondary 10B05
MathSciNet review:
0441841
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Abstract: A second order linear recurrence is a sequence of integers satisfying a where N and M are fixed integers and at least one is nonzero. If k is an integer, then the number of solutions of is at most 3 (respectively 4) if and there is an odd prime (respectively q = 3) such that and . Further is either infinite or provided that either (i) or (ii) .
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11 (1961), 833–845. MR 0136569
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R. Laxton, Linear recurrences of order two, J. Austral. Math.
Soc. 7 (1967), 108–114. MR 0207674
(34 #7489)
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J. Lewis, Diophantine equations: 𝑝adic methods,
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PrenticeHall, Englewood Cliffs, N.J.), 1969, pp. 25–75. MR 0241359
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(22 #25), http://dx.doi.org/10.1090/S00029939195901091374
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, Some diophantine problems connected with linear recurrences, Report of the Institute of the Theory of Numbers, University of Colorado, Boulder, 1959, pp. 250257.
 [1]
 R. Alter and K. K. Kubota, The diophantine equation , and a related sequence, J. Number Theory (to appear).
 [2]
 R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris 251 (1960), 12631264. MR 22 #10951. MR 0120194 (22:10951)
 [3]
 P. Chowla, S. Chowla, M. Dunton and D. J. Lewis, Some diophantine equations in quadratic number fields, Norske Vid. Selsk. Forh. 31 (1958), 181183. MR 21 #4132. MR 0105390 (21:4132)
 [4]
 S. Chowla, M. Dunton and D. J. Lewis, Linear recurrences of order two, Pacific J. Math. 11 (1961), 883845. MR 25 #39. MR 0136569 (25:39)
 [5]
 R. R. Laxton, Linear recurrences of order two, J. Austral. Math. Soc. 7 (1967), 108114. MR 34 #7489. MR 0207674 (34:7489)
 [6]
 D. J. Lewis, Diophantine equations: padic methods, Studies in Number Theory, Math. Assoc. Amer., distributed by PrenticeHall, Englewood Cliffs, N. J., 1969, pp. 2575. MR 39 #2699. MR 0241359 (39:2699)
 [7]
 A. Schinzel, The intrinsic divisors of Lehmer numbers in the case of negative discriminant, Ark. Mat. 4 (1962), 413416. MR 26 #2999. MR 0139567 (25:2999)
 [8]
 Th. Skolem, S. Chowla and D. J. Lewis, The diophantine equation and related problems, Proc. Amer. Math. Soc. 10 (1959), 663669. MR 22 #25. MR 0109137 (22:25)
 [9]
 M. Ward, Prime divisors of second order recurring sequences, Duke Math. J. 2 (1936), 472476. MR 1545940
 [10]
 , Some diophantine problems connected with linear recurrences, Report of the Institute of the Theory of Numbers, University of Colorado, Boulder, 1959, pp. 250257.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197304418412
PII:
S 00029947(1973)04418412
Keywords:
Linear recurrence,
padic numbers,
prime number,
multiplicity,
padic power series,
companion equation
Article copyright:
© Copyright 1973
American Mathematical Society
