Collisions of digit sums in bases 2 and 3

TL;DR

This paper proves a longstanding folklore conjecture that infinitely many positive integers have equal digit sums in base 2 and base 3, revealing a deep connection between these numeral systems.

Contribution

It establishes the infinite occurrence of integers with equal binary and ternary digit sums, confirming a conjecture previously considered folklore.

Findings

Infinitely many integers have equal digit sums in bases 2 and 3.

The conjecture about the equality of digit sums in these bases is proven.

The result connects properties of binary and ternary representations.

Abstract

We prove a folklore conjecture concerning the sum-of-digits functions in bases two and three: there are infinitely many positive integers $n$ such that the sum of the binary digits of $n$ equals the sum of the ternary digits of $n$.