Let \(b_3(n)\) be the number of 3-regular partitions of n. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo 2 for \(b_3(2n)\) involving every prime p with \(p \equiv 13, 17, 19, 23 \pmod {24}\) , and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo 2 for \(b_3(2n)\) involving every prime \(p\geqslant 5\) . In this paper, we introduce new infinite Ramanujan type congruences modulo 2 for \(b_3(2n)\) with cube-free period. In contrast to the work of Keith-Zanello and Yao, the congruences presented here do not originate in local obstructions and thus complement these recent results. They involve primes in \(\mathcal P=\{p \text { prime }: \exists \, j\in \{1,4,8\},\, x, y \in \mathbb Z,\, \gcd (x,y)=1 \text { with } x^2+216y^2=jp\}\) whose Dirichlet density is 1/6. As a key ingredient in our proof we show that the number of primitive solutions for \(x^2+216y^2=pm\) , \(p \in {\mathcal {P}}\) , \(p\not \mid m\) and \(pm\equiv 1\pmod {24}\) , is divisible by 8. Here, the difficulty arises from the fact that 216 is not idoneal. We also give a conjectural exact formula for the number of solutions for this Diophantine equation.
