Prime Decomposition
By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (A special case for 1 is not needed using an appropriate notion of the empty product.) However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence.
Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm. However, this is not the case with a special-purpose factorization algorithm, since it may not apply to the smaller factors that occur during decomposition, or may execute very slowly on these values. For example, if N is the number (2521 − 1) × (2607 − 1), then trial division will quickly factor 10N as 2 × 5 × N, but will not quickly factor N into its factors.
Read more about this topic: Integer Factorization Algorithms
Famous quotes containing the word prime:
“What was lost in the European cataclysm was not only the Jewish past—the whole life of a civilization—but also a major share of the Jewish future.... [ellipsis in source] It was not only the intellect of a people in its prime that was excised, but the treasure of a people in its potential.”
—Cynthia Ozick (b. 1928)