A computerassisted investigation of Ramanujan pairs
HTML articles powered by AMS MathViewer
 by Richard Blecksmith, John Brillhart and Irving Gerst PDF
 Math. Comp. 46 (1986), 731749 Request permission
Abstract:
Four new Ramanujan pairs $\{ {a_i}\}$, $\{ {b_j}\}$ are given along with the theorem that no such pairs exist with ${a_1} = 1$ and ${a_2} = s$ for any $s \geqslant 5$. 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.References

D. Acreman, Asymptotic Analysis of Partition Identities, Ph.D. Thesis, University of New South Wales, 1983.
 Henry L. Alder, The nonexistence of certain identities in the theory of partitions and compositions, Bull. Amer. Math. Soc. 54 (1948), 712–722. MR 25501, DOI 10.1090/S000299041948090620
 George E. Andrews, An incredible formula of Ramanujan, Austral. Math. Soc. Gaz. 6 (1979), no. 3, 80–89. MR 559748 R. Blecksmith, The Determination of Ramanujan Pairs, Ph.D. Thesis, University of Arizona, 1983.
 Willard G. Connor, Partition theorems related to some identities of Rogers and Watson, Trans. Amer. Math. Soc. 214 (1975), 95–111. MR 414480, DOI 10.1090/S00029947197504144809 L. Euler, Introductio in Analysin Infinitorum, Marcum Michaelem Bousquet, Lousannae, 1748, Chapter 16.
 Basil Gordon, Some continued fractions of the RogersRamanujan type, Duke Math. J. 32 (1965), 741–748. MR 184001 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1965.
 Michael D. Hirschhorn, Two further Ramanujan pairs, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 1, 1–4. MR 589461
 D. H. Lehmer, Two nonexistence theorems on partitions, Bull. Amer. Math. Soc. 52 (1946), 538–544. MR 16072, DOI 10.1090/S00029904194608605X
 J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), no. 1, 15–59. MR 501091, DOI 10.1016/00018708(78)90004X
 Thomas R. Parkin and Daniel Shanks, On the distribution of parity in the partition function, Math. Comp. 21 (1967), 466–480. MR 227126, DOI 10.1090/S00255718196702271269
 L. J. Slater, Further identities of the RogersRamanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI 10.1112/plms/s254.2.147
Additional Information
 © Copyright 1986 American Mathematical Society
 Journal: Math. Comp. 46 (1986), 731749
 MSC: Primary 11P57
 DOI: https://doi.org/10.1090/S00255718198608296439
 MathSciNet review: 829643