A short mathematical monograph about Mersenne primes—numbers of the form 2^p − 1—and the long-running effort to identify which of them are truly prime. Centered on the “32nd Mersenne prime” that falls within the set of exponents originally claimed by the 17th‑century scholar Marin Mersenne, it recounts the historical context of Mersenne’s prediction and outlines the reasoning and computations used to confirm primality, offering a concise snapshot of how conjecture, theory, and extensive calculation meet in the search for record-breaking primes.