Erdős Problem 283

References:

• erdosproblems.com/283
• [Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.

def Erdos283.Condition (p : Polynomial ℤ) : Prop

Given a polynomial p, the predicate that if the leading coefficient is positive and there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$, then for all sufficiently large $m$, there exist integers $1≤n_1<\dots < n_k$ such that $$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$ and $$m=p(n_1)+\cdots+p(n_k)$$?

Equations

• One or more equations did not get rendered due to their size.

theorem Erdos283.erdos_283 : (∀ (p : Polynomial ℤ), Condition p) ↔ sorry

Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that, for all sufficiently large $m$, there exist integers $1≤n_1<\dots < n_k$ such that $$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$ and $$m=p(n_1)+\cdots+p(n_k)$$?

theorem Erdos283.erdos_283.variants.graham : Condition Polynomial.X

Graham [Gr63] has proved this when $p(x)=x$.