The notion of broken
k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer
k, let
\(\Delta _k(n)\) denote the number of broken
k-diamond partitions of
n. Recently, Paule and Radu conjectured two relations on
\(\Delta _5(n)\) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime
p with
\(p\equiv 1\ (\mathrm{mod}\ 4)\), there exists an integer
\(\lambda (p)\in \{2,\ 3,\ 5,\ 6,\ 11\}\) such that, for
\(n, \alpha \ge 0\), if
\(p\not \mid (2n+1)\), then
$$\begin{aligned} \Delta _5\left( 11p^{\lambda (p)(\alpha +1)-1} n+\frac{11p^{\lambda (p)(\alpha +1)-1}+1}{2}\right) \equiv 0\ (\mathrm{mod}\ 11). \end{aligned}$$
Moreover, some non-standard congruences modulo 11 for
\(\Delta _5(n)\) are deduced. For example, we prove that, for
\(\alpha \ge 0\),
\(\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)\).