A computerassisted investigation of Ramanujan pairs
Authors:
Richard Blecksmith, John Brillhart and Irving Gerst
Journal:
Math. Comp. 46 (1986), 731749
MSC:
Primary 11P57
MathSciNet review:
829643
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Abstract: Four new Ramanujan pairs , are given along with the theorem that no such pairs exist with and for any . All finite Ramanujan pairs are determined and their significance in bounding the local branching degree in the search tree for such pairs is discussed. The search techniques and programs that were used are also described. The parity of the coefficients in the power series is determined in two of the new identities. Partition interpretations of the six recent identities are also given.
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 D. Acreman, Asymptotic Analysis of Partition Identities, Ph.D. Thesis, University of New South Wales, 1983.
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 H. L. Alder, "The nonexistence of certain identities in the theory of partitions and compositions," Bull. Amer. Math. Soc., v. 54, 1948, pp. 712722. MR 0025501 (10:16c)
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 G. E. Andrews, "An incredible formula of Ramanujan," Austral. Math. Soc. Gaz., v. 6, 1979, pp. 8089. MR 559748 (81d:10009)
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 R. Blecksmith, The Determination of Ramanujan Pairs, Ph.D. Thesis, University of Arizona, 1983.
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 W. Connor, "Partition theorems related to some identities of Rogers and Watson," Trans. Amer. Math. Soc., v. 214, 1975, pp. 95111. MR 0414480 (54:2581)
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 L. Euler, Introductio in Analysin Infinitorum, Marcum Michaelem Bousquet, Lousannae, 1748, Chapter 16.
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 B. Gordon, "Some continued fractions of the RogersRamanujan type," Duke Math. J., v. 32, 1965, pp. 741748. MR 0184001 (32:1477)
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 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1965.
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 T. R. Parkin & D. Shanks, "On the distribution of parity in the partition function," Math. Comp., v. 21, 1969, pp. 466480. MR 0227126 (37:2711)
 [13]
 L. J. Slater, "Further identities of the RogersRamanujan type," Proc London Math. Soc. (2), v. 54, 1952, pp. 147167. MR 0049225 (14:138e)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608296439
PII:
S 00255718(1986)08296439
Keywords:
Ramanujan pairs,
computer search tree
Article copyright:
© Copyright 1986
American Mathematical Society
