Factorization

Enter a signed integer; decimal and 0x/0b/0o notation are accepted. Zero has no finite prime factorization. Negative values include −1; arithmetic functions use |n|. φ counts coprime positive residues, τ counts positive divisors and σ sums them. For 1, all three equal 1.

Work stays in the browser and can be cancelled. A searching indicator shows the current phase; no meaningful completion percentage is known. If a work limit is reached, a bracketed cofactor is explicitly unfinished. The search has a 15-second time limit and bounded divisor operations.

Prime testing is deterministic below 318665857834031151167461; beyond that bound, passing factors are only probable primes. Arithmetic functions are shown only for a certified factorization. Large cryptographic composites may remain unresolved. Input limit: 512 characters.

Under Integer lists, spaces/commas/semicolons separate 1–128 signed integers. Gcd and lcm are nonnegative, gcd(0,0)=0 and any zero makes lcm zero. The displayed Bézout identity reconstructs the gcd from the original signed numbers.