Number Theory
Algorithms for number theoretical calculations are studied in computational number theory.
| Operation | Input | Output | Algorithm | Complexity |
|---|---|---|---|---|
| Greatest common divisor | Two n-digit numbers | One number with at most n digits | Euclidean algorithm | O(n2) |
| Binary GCD algorithm | O(n2) | |||
| Left/right k-ary binary GCD algorithm | O(n2/ log n) | |||
| Stehlé–Zimmermann algorithm | O(log n M(n)) | |||
| Schönhage controlled Euclidean descent algorithm | O(log n M(n)) | |||
| Jacobi symbol | Two n-digit numbers | 0, −1, or 1 | ||
| Schönhage controlled Euclidean descent algorithm | O(log n M(n)) | |||
| Stehlé–Zimmermann algorithm | O(log n M(n)) | |||
| Factorial | A fixed-size number m | One O(m log m)-digit number | Bottom-up multiplication | O(m2 log m) |
| Binary splitting | O(log m M(m log m)) | |||
| Exponentiation of the prime factors of m | O(log log m M(m log m)), O(M(m log m)) |
Read more about this topic: Computational Complexity Of Mathematical Operations
Famous quotes containing the words number and/or theory:
“... is it not clear that to give to such women as desire it and can devote themselves to literary and scientific pursuits all the advantages enjoyed by men of the same class will lessen essentially the number of thoughtless, idle, vain and frivolous women and thus secure the [sic] society the services of those who now hang as dead weight?”
—Sarah M. Grimke (1792–1873)
“The great tragedy of science—the slaying of a beautiful theory by an ugly fact.”
—Thomas Henry Huxley (1825–1895)