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smooth number
In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman.〔M. E. Hellman, J. M. Reyneri, "Fast computation of discrete logarithms in GF (q)", in ''Advances in Cryptology: Proceedings of CRYPTO '82'' (eds. D. Chaum, R. Rivest, A. Sherman), New York: Plenum Press, 1983, p. 3–13, (online quote ) at Google Scholar: "Adleman refers to integers which factor completely into small primes as “smooth” numbers."〕 Smooth numbers are especially important in cryptography relying on factorization. The 2-smooth numbers are just the powers of 2.
==Definition==
A positive integer is called B-smooth if none of its prime factors is greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, despite the fact that they miss out prime factors 3 and 5 respectively. 5-smooth numbers are also called ''regular numbers'' or ''Hamming numbers''; 7-smooth numbers are also called ''humble'', and sometimes called ''highly composite'', although this conflicts with another meaning of highly composite numbers.
Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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