Cyclotomic numbers and Jacobi sums, introduced over two centuries ago by Gauss and Jacobi, respectively, are pivotal in number theory and find wide applications in combinatorial designs, coding theory, cryptography, and information theory. The cyclotomic problem, focused on determining all cyclotomic numbers, or equivalently evaluating all Jacobi sums of a given order, has been a subject of extensive research. This paper explores their trivariate counterparts, termed "ternary cyclotomic numbers" and "ternary Jacobi sums", highlighting the fundamental properties that mirror those of the classical cases. We show the ternary versions of Fourier series expansions, two symmetry properties, and a summation equation. We further demonstrate that ternary Jacobi sums, with at least one trivial variable, can be evaluated in terms of classical Jacobi sums of the same order. These properties are established through elementary methods that parallel those utilized in classical cases. Based on these properties, then we offer explicit calculations for all ternary Jacobi sums and ternary cyclotomic numbers of order $e=2$, and near-complete results for order $e=3$, with the exception of the elusive integer $J_3(1,1,2)$ for us.

This work establishes a new probabilistic bound on the number of elements needed to generate finite nilpotent groups. Let ${\phi }_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0<ϵ<1$, we prove that ${\phi }_k(G)\ge 1-ϵ$ if $k\ge \mathrm{rank}(G)+\lceil {\log }_2(2/ϵ)\rceil$ (a bound based on the group rank) or if $k\ge \mathrm{len}(G)+\lceil {\log }_2(1/ϵ)\rceil$ (a bound based on the composition length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of $k\ge \lceil {\log }_2|G|+{\log }_2(1/ϵ)\rceil +2$. Our results provide a foundational tool to analyze probabilistic algorithms, thereby enabling a better estimation of the iteration count for the finite abelian hidden subgroup problem (AHSP) standard quantum algorithm and a reduction in the circuit repetitions required by Regev's factoring algorithm.