Godara–Sarkar conjecture for the small Davenport constant of Heisenberg groups

Godara and Sarkar conjectured the formula $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$ and proved the case $p=3$. The general conjecture has now attracted claims covering the next two cases and all primes.

Known results

• Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ ($p=3$).

August 2026 claimed uniform proof

A later preprint claims a theoretical proof of $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$. It presents the $p=5$ and $p=7$ results of White and Volkmann as earlier special cases and says its uniform argument removes the need for finite computation. The claim is unverified. White’s July preprint claimed $\mathsf{d}(H_{125})=12$, and Volkmann’s August preprint claimed $\mathsf{d}(H_{343})=18$, both matching the conjecture.

Current status (as of August 2026): $p=3$, $p=5$, and $p=7$ have claimed proofs, and a preprint claims the all-odd-prime formula, but independent verification is absent and the conjecture is not yet settled.