Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Smooth numbers: computational number theory and beyond
Andrew Granville, Smooth numbers: computational number theory and beyond, in Algorithmic Number Theory, MSRI Publications 44 (2008), beginning on p. 267. The selected text is the chapter PDF supplied by the SLMath library, with its own printed pagination.
No explicit copyright notice or redistribution license was found in the selected PDF, inspected 2026-10-15. Public access does not establish redistribution rights. No file of this source is held.
Scope
The chapter surveys smooth-number counting estimates, applications to factoring and other algorithms, and methods for intervals, arithmetic progressions and polynomial values. Section 4.1, printed pp. 294–295, distinguishes uniform short-interval results, almost-all results and conjectures, and points to Friedlander–Lagarias, Harman and Croot.
The range summary for Harman on p. 295 is not used here as an unconditional replacement for the original theorem. Harman's Theorem 2 requires its exponent-pair condition and makes the simpler formula for conditional on the exponent-pair conjecture; see the original statement and its qualifications.
Read status: claims checked. The abstract, introductory scope and Section 4.1 were read; p. 295 was inspected as a page image for the cited range summary. The bibliography entries for Friedlander–Lagarias and Harman were checked. No proof sketch in the chapter, or proof in its cited sources, is verified by this reading.
Bears on. Problem 962: Section 4.1 locates relevant smooth-integer results and separates the uniform question from weaker distribution statements. The exact original hypotheses remain the basis for any application to the problem.