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arXiv · 2609.37301

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

Soon-Yi Kang
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Abstract

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 $Γ_0(2)$ 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 $n\ge2$. 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 $B_1$.

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Soon-Yi Kang. 2026-09-29. The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression. https://arxiv.org/abs/2609.37301

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