Editing Smooth number (section)

==Smooth over a set ''A''==
Moreover, ''m'' is said to be smooth over a [[Set (mathematics)|set]] ''A'' if there exists a factorization of ''m'' where the factors are powers of elements in ''A''. For example, since 12 = 4&thinsp;×&thinsp;3, 12 is smooth over the sets ''A''<sub>1</sub> = {4,&thinsp;3}, ''A''<sub>2</sub> = {2,&thinsp;3}, and <math>\mathbb{Z}</math>, however it would not be smooth over the set ''A''<sub>3</sub> = {3,&thinsp;5}, as 12 contains the factor 4 = 2<sup>2</sup>, and neither 4 nor 2 are in ''A''<sub>3</sub>.

Note the set ''A'' does not have to be a set of prime factors, but it is typically a proper [[subset]] of the primes as seen in the [[factor base]] of [[Dixon's factorization method]] and the [[quadratic sieve]]. Likewise, it is what the [[general number field sieve]] uses to build its notion of smoothness, under the [[homomorphism]] <math>\varphi:\mathbb{Z}[\theta]\to\mathbb{Z}/n\mathbb{Z}</math>.<ref>{{cite web|url=https://personal.math.vt.edu/brown/doc/briggs_gnfs_thesis.pdf|title=An Introduction to the General Number Field Sieve|first=Matthew E.|last=Briggs|publisher=Virginia Polytechnic Institute and State University|website=math.vt.edu|date=17 April 1998|location=Blacksburg, Virginia|access-date=26 July 2017}}</ref>