Primitive Root Modulo Prime Calculator

Factor p minus one into distinct primes and test candidates until every reduced-order power differs from one modulo p. Prime residue groups are cyclic, while arbitrary composite moduli need different existence checks. Inputs remain whole numbers inside the displayed domain, and the page never rounds a decimal into an accepted integer. This keeps the reported result tied to one explicit mathematical definition rather than an unstated convention.

The result is the smallest positive generator found; other primitive roots can generate the same nonzero residue group. Use the factor tests shown, then confirm the power at p minus one returns to one. Reuse the displayed recurrence, congruence, or counting identity with the same inputs. The verification evidence checks this calculator's result, but it does not establish assumptions or conclusions outside the stated mathematical domain.

The prime cap keeps factorization and candidate search predictable; the result is not approval of a cryptographic group. Larger moduli require more capable factorization and deliberate security review. A finite limit prevents browser stalls and avoids presenting an unsafe floating-point integer as exact. Work beyond that limit belongs in arbitrary-precision software with explicit resource controls and independent validation.