A computer-assisted investigation of Ramanujan pairs

Abstract

Abstract. Four new Ramanujan pairs {a,}, {6,} are given along with the theorem that no such pairs exist with at = 1 and a2 = s for any í > 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.

Some Infinite Product Identities

“…An Expansion Formula. We now establish a useful expansion formula for the product of two general T-functions defined in (2) Proof. Let…”

“…Applying Theorem 2 with ki -2, ¿i = 0, pi = 2 and k2 = l2 = p2 = 1, the separability condition is satisfied with a = 2, 6 = 1, and m = 3. Taking R = {0,1, -1}, we find, after some simplification, that r1(2,0;2)T1(l,l;l) = r1 (6,2)[To(3,l;3)-2;-1ro(3,-l;3)] oo = (o)4 y x3n2+n(z3n -z-3"-1).…”

“…3. Formula (29) gives an immediate proof of Theorem 1 in [6] and identity (2) in [6], viz., with m = 2, a = b = 1, and R = {0,1}, we have T02(1,0;1) = ^a;'-%tTo(2,-2r;0)To(2,2r;2) r€R oo oo oo oo = Ei2"2 E z2"2*2"+xz E *2n2+2n E *2n2+2nz2n- This search algorithm relies on the function DeadendDegree, which attempts to factor the series 1 + J2n=i anXn (stored as an array (<*i,..., oil)) hito a product of the form on the left-hand side of (45) up to the specified degree L. If no factorization is possible, the program returns the degree at which a contradiction is first obtained. Otherwise, the value L + 1 is returned.…”

Parity Results for Certain Partition Functions and Identities Similar to Theta Function Identities

“…In [1] two such results were established, namely: (1) T[(l-x"y1=txe(n)(mod2) «SS -or, for S = {« e N: « = +(2,3,4,5,6,7) (mod20)}, e(n) = n(5n + l)/2 and 5 = {« g N: « = +(1,2,5,6,8,9) (mod20)}, e(n) = n(5n + 3)/2 respectively. (See entries 4 and 5 in Table 1 below.)…”

“…(See entries 4 and 5 in Table 1 below.) In this paper we present a collection of parity results (obtained by using the methods discussed in [1]), which have been discovered over the last two years on an IBM Personal Computer and the Data General Eclipse S/230 at the Mathematics Department of the University of Arizona (see [2]). …”