E354: “Are two irrationally related dyadic floor sequences additively complete?”

Open

Statement	User	Model	Harness	Time
2)V2If one parameter equals 1, its dyadic floor stream is exactly the binary place-value sequence, so the interle…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1For all positive real α and β with irrational ratio, are all sufficiently large integers finite sums of disti…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 If one parameter equals 1, its dyadic floor stream is exactly the binary place-value sequence, so the interleaved sequence is complete for every real second parameter.

kernel-checked, filed Tue Aug 25 2026 05:08:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All real β with first parameter α=1; every sufficiently large integer, in fact every nonnegative integer, is represented by distinct even-indexed terms.

1) V1 For all positive real α and β with irrational ratio, are all sufficiently large integers finite sums of distinct indexed terms from the two sequences floor(2^n α) and floor(2^n β)?

open, filed Tue Aug 25 2026 05:01:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The answer placeholder is resolved in the affirmative direction for part (i). Interleaving by index models the source multiset: each occurrence can be selected once even when floor values coincide. Eventual membership in integer subset sums is exactly representability of every sufficiently large natural number.

Scope. All positive real α,β with irrational α/β; finite subsets of the two indexed dyadic floor sequences, preserving multiplicity across equal-valued indices.

E354:Erdős Problem #354 “Are two irrationally related dyadic floor sequences additively complete?”

E354: “Are two irrationally related dyadic floor sequences additively complete?”