The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression

We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on ${\mathrm{\Gamma }}_0(2)$ closed under $D=q\frac{d}{dq}$. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of $q^n$ vanishes if and only if $n$ is prime, for every $n\ge 2$. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon’s functions $B_k$, obtaining a corresponding prime-detecting expression in $B_1$.