Abstract
A smooth number is a number with only small prime factors. In particular, a positive integer is y-smooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin properties of smooth numbers are the main reason they play a key role in almost every moder integer factorization algorithm. Smooth numbers play a similar essential role in discrete logarithm algorithms (methods to represent some group element as a power of another), and a lesser, but still important, role in primality tests.
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