2025/01/01 by Michael Drmota, Drmota, Michael, Lukas Spiegelhofer +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary: 11A63 #Secondary: 11N60
paper · pdf · doi:10.48550/arxiv.2501.00850
openalex publication_date 2025/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider the sum-of-digits functions s2 and s3 in bases 2 and 3. These functions just return the minimal numbers of powers of two (resp. three) needed in order to represent a nonnegative integer as their sum. A result of the second author states that there are infinitely many collisions of s2 and s3, that is, positive integers n such that s2(n)=s3(n). This resolved a long-standing folklore conjecture. In the present paper, we prove a strong generalization of this statement, stating that (s2(n),s3(n)) attains almost all values in \mathbb N2, in the sense of asymptotic density. In particular, this yields generalized collisions: for any pair (a,b) of positive integers, the equation as2(n)=bs3(n) admits infinitely many solutions in n.