Research Article
Proof of Some Conjectures of Melham Using
Ramanujan’s
1
1
Formula
Bipul Kumar Sarmah
Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam 784028, India
Correspondence should be addressed to Bipul Kumar Sarmah; bipul@tezu.ernet.in
Received 5 February 2014; Accepted 16 June 2014; Published 10 July 2014
Academic Editor: Wolfgang zu Castell
Copyright © 2014 Bipul Kumar Sarmah. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We employ Ramanujan’s
1
1
formula to prove three conjectures of R. S. Melham on representation of an integer as sums of
polygonal numbers.
1. Introduction
Jacobi’s classical two-square theorem is as follows.
Teorem 1 (see [1]). Let {◻ + ◻}() denote the number of
representations of as a sum of two squares, counting order
and sign, and let
,
() denote the number of positive divisors
of congruent to modulo . Ten
{◻ + ◻} () = 4 (
1,4
() −
3,4
()). (1)
Te above theorem can also be recasted in terms of
Lambert series as
∞
∑
=0
{◻ + ◻} ()
=1+4
∞
∑
=0
(
4+1
1−
4+1
−
4+3
1−
4+3
).
(2)
Similar representation theorems involving squares and tri-
angular numbers were found by Dirichlet [2], Lorenz [3],
Legendre [4], and Berndt [5]. For example, the following two
theorems are due to Lorenz and Ramanujan, respectively.
Teorem 2 (see [3]). Let {◻ + ◻}() denote the number of
representations of as a sum of times a square and times a
square. Ten
{◻ + 3◻} () = 2 (
1,3
() −
2,3
())
+ 4 (
4,12
() −
8,12
()).
(3)
Teorem 3 (see [5]). Let {Δ + Δ}() denote the number of
representations of as a sum of times a triangular number
and times a triangular number. Ten
{Δ + 3Δ} () =
1,3
(2 + 1) −
2,3
(2 + 1) . (4)
Hirschhorn [6, 7] obtained forty-fve similar identities
(including those obtained by Legendre and Ramanujan)
involving squares, triangular numbers, pentagonal numbers,
and octagonal numbers employing dissection of the -
series representations of the identities obtained by Jacobi,
Dirichlet, and Lorenz. In [8], Baruah and the author obtained
twenty-fve more such identities involving squares, triangular
numbers, pentagonal numbers, heptagonal numbers, octag-
onal numbers, decagonal numbers, hendecagonal numbers,
dodecagonal numbers, and octadecagonal numbers. More
works on this topic have been done in [9–11]. In [11], Melham
presented 21 conjectured analogues of Jacobi’s two-square
theorem which are verifed using computer algorithms. In
[12], Toh ofered a uniform approach to prove these conjec-
tures using known formulae for {◻ + ◻}(). In this paper,
we show that some of these conjectures can also be proved
by using Ramanujan’s famous
1
1
formula. We prove three
conjectures enlisted in the following theorem which have
appeared as (6), (7), and (8), respectively, in [11].
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2014, Article ID 738948, 6 pages
http://dx.doi.org/10.1155/2014/738948