# nt.number theory – Congruences for “colored partitions” a la Ramanujan

Yes, these are all true and are all in the literature. The first two congruences are part of infinite families of the form
$$begin{cases} p_{ell-1}(ell n+a)equiv 0mod ell, \ p_{ell-3}(ell n+b)equiv0mod ell, end{cases}$$
where $$ell geq 5$$ is a prime and $$a,b$$ are such that: $$24a+1$$ is a quadratic nonresidue, $$8b+1$$ is either a quadratic nonresidue or divisible by $$ell$$. Both congruences are immediate consequences of the Euler pentagonal theorem and Jacobi triple product respectively. This is explained in section 2 of the paper: I. Kiming and J. Olsson, Congruences like Ramanujan’s for powers of the partition function (Arch. Math., 59(4):348-360, 1992) doi: 10.1007/BF01197051.

The third identity is a bit more recent. It was proven in Lin, B.L.S. Ramanujan-style proof of $$p_{−3}(11n+7)≡0 pmod{11}$$ (Ramanujan J 42, 223–231 (2017)) doi:10.1007/s11139-015-9733-5. It was generalized to the following infinite family
$$p_{ell-8}left(ell n+frac{ell^2-1}{3}-ell lfloorfrac{ell^2-1}{3ell}rfloorright)equiv 0 mod ell.$$
For this last congruence and many other “exceptional” infinite families see the paper Locus, M., Wagner, I. Congruences for Powers of the Partition Function (Ann. Comb. 21, 83–93 (2017)) doi:https://doi.org/10.1007/s00026-017-0342-4.