Jacobi Symbol Integer Calculator

Normalize a modulo positive odd n, remove factors of two, and repeatedly apply quadratic reciprocity until the reduced numerator is zero. The denominator may be composite, which is the key distinction from a Legendre symbol. 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 zero value proves a common factor, while a value of one does not guarantee a square root when the denominator is composite. For an odd prime denominator, Euler's criterion supplies a separate residue check. 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 denominator is capped at 1,000,001 for predictable browser work; this is educational number theory, not a cryptographic decision service. The implementation favors transparent integer reductions over unbounded factorization. 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.