Quick guide
How to use this calculator
- Choose the required modular or symbol mode when one is available.
- Enter the labelled base-10 integers or comma-separated integer lists.
- Read the exact classification, residue, factorization, solution family, or bounded sequence.
Calculation method
Apply the exact integer algorithm
Merge congruences with extended Euclid; compatibility requires each residue difference be divisible by the modulus GCF.
The engine uses exact BigInt arithmetic and explicit work limits. Undefined inverses, incompatible congruences, invalid prime assumptions, and unsupported exhaustive searches return an error or a mathematically distinct no-solution result.
Worked example
Worked example
x≡2 mod 3 and x≡3 mod 5 gives x≡8 mod 15.
Merge congruences with extended Euclid; compatibility requires each residue difference be divisible by the modulus GCF.
Supported inputs
Precision and limits
Exact arithmetic
Integer arithmetic and displayed residues are exact. No floating-point conversion is used in the calculation engine.
Work limits
Factorization and divisor tools support positive inputs through 10¹². Enumeration tools label limits of 100,000 or 1,000,000; deterministic primality testing supports n<2⁶⁴. General non-enumerated integer fields accept up to 100 digits.
Conventions
Modulo results use the least non-negative residue. Divisors are positive unless a theorem explicitly uses signed or directed integers. Primality begins at 2.
Scope
Exploratory calculators compute only the entered bounded case. A Goldbach pair, Carmichael classification, or other result does not claim to settle a general conjecture beyond that input.