A computer-assisted investigation of Ramanujan pairs

Blecksmith, Richard; Brillhart, John; Gerst, Irving

doi:10.1090/S0025-5718-1986-0829643-9

A computer-assisted investigation of Ramanujan pairs
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by Richard Blecksmith, John Brillhart and Irving Gerst PDF

Math. Comp. 46 (1986), 731-749 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

Asymptotic Analysis of Partition Identities

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/S0002-9904-1948-09062-0
George E. Andrews, An incredible formula of Ramanujan, Austral. Math. Soc. Gaz. 6 (1979), no. 3, 80–89. MR 559748

The Determination of Ramanujan Pairs

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/S0002-9947-1975-0414480-9

Introductio in Analysin Infinitorum

Basil Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741–748. MR 184001

An Introduction to the Theory of Numbers

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/S0002-9904-1946-08605-X
J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), no. 1, 15–59. MR 501091, DOI 10.1016/0001-8708(78)90004-X
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/S0025-5718-1967-0227126-9
L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI 10.1112/plms/s2-54.2.147