Z-Number

A -number is a real number such that

for all , 2, ..., where is the fractional part of . Mahler (1968) showed that there is at most one -number in each interval for integer , and therefore concluded that it is unlikely that any -numbers exist. The -numbers arise in the analysis of the Collatz conjecture.

More generally, for relatively prime positive integers , let denote the set of positive real numbers for which the fractional parts lie in the interval for every nonnegative integer . Stephan (2026) reports

Stephan (2026) compares this excluded interval, whose length is , with an interval of length excluded by Dubickas (2019). This comparison concerns the length of a single excluded interval, not the best bound obtained from the nearest integer function, and it does not settle Mahler's original problem . VibeMathed (2026) reports that Fable 5 and Opus 5 discovered and formalized the result and drafted the manuscript from the Lean development. The prior-art audit accompanying Stephan (2026) records the novelty as unknown. Independent specialist review of the proof had not been reported (VibeMathed 2026).