On Diophantine equations over the ring of all algebraic integers

For each odd prime q an integer NH_q (NH₃ = ?1, NH₅ = ?1, NH₇ = 97, NH₁₁ = ?243, …) is defined as the norm from L to $\text{Q}$ of the Heilbronn sum H_q = Tr_I^{$\text{Q}$(ζ)}(ζ), where ζ is a primitive q²th root of unity and L ?- $\text{Q}$(ζ) the subfield of degree q. Various properties are proved relating the congruence properties of H_q and NH_q modulo p (p ≠ q prime) to the Fermat quotient $\text{(p}{}^{\mathrm{q\; ?\; 1}}\text{? 1)}\text{q}\text{(}\text{mod}\text{q)}$; in particular, it is shown that NH_q is even iff 2^{q ? 1} ≡ 1 (mod q²).