# Deleting the powers of four

Open problem settled, verified by computation. Vamshi Jandhyala.

> Remove 16, 64, 256, … and the ordered ways to write n as a sum of three survivors still rise at every n: Sándor and Yang's Remark 1.3.

Canonical: https://vamshij.com/research/powers-of-four-representations
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/powers-of-four-representations
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/powers-of-four-representations/paper/note.pdf

Let `R_{A,h}(n)` count the ordered `h`-tuples from a set `A` of natural numbers that add up to `n`. Sándor and Yang ([arXiv:2606.28849](https://arxiv.org/abs/2606.28849), Remark 1.3) asked whether `R_{A,h}` is strictly increasing for `A = ℕ ∖ {4^(j+2) : j ≥ 0}`. It is, for every `h ≥ 3` and from `n = 0` on (Proposition 1).

Bell and Shallit's Theorem 1 ([arXiv:2212.12473](https://arxiv.org/abs/2212.12473)) already gives strict increase for all large `n`, and the argument is the coefficient computation of their Theorem 2. What the note adds is the explicit bound `R_{A,3}(N) − R_{A,3}(N−1) ≥ N + 1 − 3m(N) − m(N)²` (Lemma 2), valid for every deleted set, with `m(N)` the number of deleted elements up to `N`, and the reduction from every `h ≥ 3` to `h = 3` (Lemma 3). It also records two facts about Sándor and Yang's Problem 1.4, which asks which densities below 1 are possible: no eventually periodic set of density strictly between 0 and 1 works (Proposition 4), and any set that works has `A(N)³ ≥ (N+1)(N+2)/2` (Proposition 5). A density-3/4 candidate passes an exact check up to 262,144 but is unproved (Remark 6).

Preprint v1, 9 October 2026, not peer reviewed. Lemmas 2 and 3 and Proposition 1 are formally verified in Lean 4 (`lean/`, standard axioms only); Propositions 4 and 5 are proved by hand. The author used AI tools (Codex, OpenAI; Claude, Anthropic) in this work, as described in the paper's acknowledgement, and is responsible for its content.