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
Cross out multiples beginning at p² for each remaining p≤√n.
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
Through 20 the primes are 2, 3, 5, 7, 11, 13, 17, 19.
Cross out multiples beginning at p² for each remaining p≤√n.
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.
Calculator-specific rule
The inclusive list bound is 100,000 so the full prime output stays usable; Prime Counting supports 1,000,000 without listing every prime.