Problem 407

Let $w(n)$ count the number of solutions to

n=2^a+3^b+2^c3^d

with $a,b,c,d\geq 0$ integers. Is it true that $w(n)$ is bounded by some absolute constant?

Source: erdosproblems.com/407

Number theory

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SolvedProved

An accepted solution exists. The statement is true.

Proved: the site's label; its commentary credits Evertse, Győry, Stewart and Tijdeman with the proof. The frontmatter standing is derived from the accepted claim pages their proof, Tijdeman and Wang's bound of four and Bajpai and Bennett's effective bounds, each accepted on the site's credit and, for the two journal papers, on their refereed publication.

BaBe24: Bajpai, Prajeet and Bennett, Michael A., Effective $S$ -unit equations beyond three terms: Newman's conjecture. Acta Arith. (2024), 421-458.
EGST88: Evertse, J.-H. and Győry, K. and Stewart, C. L. and Tijdeman, R., $S$ -unit equations and their applications. New advances in transcendence theory (Durham, 1986) (1988), 110-174.
TiWa88: Tijdeman, R. and Wang, Lian Xiang, Sums of products of powers of given prime numbers. Pacific J. Math. (1988), 177-193.

Problem 407

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