Modular Square Root Calculator

Validate an odd prime modulus, test quadratic residuosity, then use Tonelli-Shanks and normalize both roots. Composite moduli can require factorization and a different solution structure, so they are rejected. 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.

A nonzero quadratic residue has two roots modulo an odd prime; zero has one distinct root and a nonresidue has none. Square each displayed root and reduce the product modulo p to recover the normalized target. 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.

Prime moduli are capped at 1,000,003; the route does not solve composite-modulus roots or certify cryptographic parameters. The cap bounds primality checks and Tonelli-Shanks modular operations. 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.