<p>We study the problem of exact equation discovery, i.e., identifying symbolic equations that perfectly describe relationships in noise-free data. While most existing approaches focus on approximate recovery from noisy measurements, we consider settings in which exact correctness is required. This setting is closely related to methods that infer symbolic relations, such as recurrence equations or generating functions, from finite data. We show that exact equation discovery can be formulated as the computation of the vanishing ideal of the observed data and leverage Gröbner bases as an effective algorithmic tool. Building on this connection, we introduce MoadeeB, a new algorithm for discovering exact equations over integers and rational numbers. We evaluate MoadeeB in a large-scale empirical study on more than 30,000 integer sequences from the Online Encyclopedia of Integer Sequences (OEIS), focusing on the reconstruction of known recurrences and the discovery of previously undocumented ones. We compare against state-of-the-art symbolic regression and program synthesis approaches, as well as approaches from experimental mathematics and computer algebra that infer symbolic relations directly from finite sequence prefixes. The results show that MoadeeB achieves competitive or superior performance across these method classes, while additionally enabling the discovery of exact equations beyond the scope of existing approaches.</p>