Multiplicative Order Modulo Calculator

Validate coprimality, normalize the base, and iterate exact modular products up to the Carmichael bound until residue one first appears. If gcd(a,m) exceeds one, a is not a unit and this multiplicative order is undefined. Inputs must be whole numbers inside the displayed safe bound. The calculator does not round decimals into the domain or silently substitute a different convention, so verify each entered integer before interpreting the exact result.

The order describes the cycle length of one invertible residue, not the universal cycle length of every residue. Computing a^order modulo m must return one, while every smaller positive exponent must fail that test. Use the displayed factors, recurrence step, or modular residue as an independent check. Exact integer output means the implemented arithmetic has no decimal approximation within its bound; it does not prove an unrelated theorem or an unstated combinatorial model.

This educational repeated-product implementation is bounded and is not a discrete-logarithm solver or cryptographic primitive. Large-modulus order finding generally uses factorization and stronger algorithms rather than an unrestricted linear search. The bound keeps browser work predictable and every returned integer within JavaScript's exact safe range. Larger inputs need arbitrary-precision software, explicit resource controls, and independent verification rather than treating an overflowed floating-point value as exact.