bibkey: golomb1970powerful authors: Solomon W. Golomb year: 1970 title: Powerful numbers doi: 10.1080/00029890.1970.11992654 claim: Every powerful number is the product of a perfect square and a perfect cube, and the paper studies the distribution of such numbers. strata_touched:

• D5/S3/Arith/Powerful/PowerfulNumber license: citation-only triage: anchor

Powerful numbers

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• DOI: https://doi.org/10.1080/00029890.1970.11992654
• American Mathematical Monthly 77 (1970), 848-852.

Scope

Golomb defines a positive integer to be powerful when every prime dividing it has its square dividing it, and records that such an integer can be written as a square times a cube. The term “powerful number” originates in this paper. Powerful numbers appear in several Erdős problems, among them the question of whether three consecutive powerful numbers exist.

The Lean module states the representation in the form: for n at least one and powerful, there exist a and b with n equal to a squared times b cubed and b squarefree. The squarefree part b is uniquely determined, although uniqueness is not stated.

The proof is a repository derivation from Mathlib’s square-times-squarefree decomposition. Writing n as d squared times c with c squarefree, the powerful condition forces every prime of c to occur in n to order at least two while occurring in c to order exactly one, hence to divide d; comparing factorization exponents gives c dividing d, and substituting d equal to e times c yields the displayed representation with a equal to e and b equal to c. No distribution or counting result of the source is formalized.