Since K. K. Norton’s survey [Mem. Am. Math. Soc. 106, 106 p. (1971; Zbl 0211.37801)], there has been an explosion of interest in the distribution of integers without large prime factors, the so-called “smooth numbers”, in large part because of the ground-breaking contributions of the present two authors, Alladi, Balog, Erdős, Friedlander, Hensley, Ivić, Pomerance and many others. Asymptotic estimates have now been proved, in very wide ranges, for the number of smooth numbers up to \(x\), in arithmetic progressions, in short intervals, as one less than a prime, \(\dots\). Moreover they have been applied to help our understanding of a wide diversity of number theory questions, such as Waring’s problem, the Fermat and \(abc\) conjectures, the analysis of algorithms, particularly for factoring and primality testing, Carmichael numbers, gaps between primes, and a host of other problems besides. The authors here survey these and other developments in the subject.
The survey is excellently organized, well-written, thorough, and goes into considerable depth for those questions which deserve that kind of attention. This will now complement Norton’s survey as the primary references in the area; and it also will serve as an excellent guide for those who wish to learn about the subject.