On the {0,1} Case of Guy's Problem F24: Reduction to a Pell‑Type Obstruction with Leading‑Digit Constraints

Abstract

We study positive integers n such that the decimal expansion of n²,without leading zeros,uses only the digits 0 and 1.The known examples are the powers of 10.We prove a structural reduction to a Pell‑type family and a leading‑digit restriction to {1,3}.First,via a p‑adic valuation analysis,any counterexample must belong to one of the two Pell‑type families 5^{t+2}w−2^{t}u=±1, uw∈{0,1}.All other families are excluded by elementary decimal‑digit arguments.Second,using a simple leading‑digit analysis,we prove that if n² ∈ {0,1} and n is not a power of 10,then the leading digit of n is either 1 or 3.Combining these two reductions,the remaining obstruction is that the Pell‑generated integer n=50·10^{t}k+B, B²−4(25·10^{t})A=1,cannot have leading digit 1 or 3 and satisfy n² ∈ {0,1}*.We state this remaining problem as an open proposition.The conjecture remains open.