Does $\{p+\lfloor C^k\rfloor\}$ have positive density for every $C>1$? (Erdős #244)
Statement
Fix a real number $C>1$. Consider the set of all integers of the form $p+\lfloor C^k\rfloor$, where $p$ ranges over primes and $k\ge 0$ over nonnegative integers (here $\lfloor\cdot\rfloor$ is the floor function). Does this set always have positive natural (lower) density in the integers, for every $C>1$?
Acceptance. FULLY RESOLVES: a complete proof that for every $C>1$ the set $\{p+\lfloor C^k\rfloor\}$ has positive lower density — machine-checkable (Lean/Coq) preferred, otherwise a full written proof; OR a disproof exhibiting a specific $C>1$ for which the density is zero, with proof. ADVANCES (each a complete proof, strictly beyond the background frontier — note Ding [Di25] already covers almost all $C$): prove positive density for a specific individual $C>1$ not already covered by prior work (e.g. a named transcendental or algebraic value); OR enlarge the class of admissible $C$ beyond the measure-zero-exceptional barrier by proving it for an explicitly described family (e.g. all $C$ satisfying a stated Diophantine or transcendence condition); OR establish a quantitative positive-density lower bound uniform over a specified family of $C$. Deliver a machine-checkable or complete written proof.
Background
Posed to Erdős by Kalmár and recorded by Erdős [Er61, p.230]; listed as open on erdosproblems.com/244 (fetched 2026-07-21, status 'open'). Erdős believed the answer is yes. For integer $C$ this is a classical theorem of Romanoff [Ro34], who showed the integers of the form $p+C^k$ have positive density (the case $C=2$, primes plus powers of two, being the original). There is substantial recent progress: Ding [Di25] proved the positive-density statement for almost all $C>1$ (all $C$ outside a set of Lebesgue measure zero). What remains open is the full statement — every $C>1$, and in particular explicit individual values of $C$ (e.g. specific transcendental or algebraic $C$) that might lie in the measure-zero exceptional set left by Ding's theorem. The statement has a Lean formalization in the formal-conjectures project; no Erdős prize is attached. Attacker's tool: Romanoff-type arguments (Cauchy–Schwarz on the representation function together with bounds on the associated reciprocal sums) adapted to the floor sequence $\lfloor C^k\rfloor$; numerical estimation of the density for specific $C$ can guide the search for a genuine exceptional value or corroborate universality.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #244 (T. F. Bloom) | website |
| REF-02 | Lean formalisation (formal-conjectures, Erdős #244) | website |
Investigations · 0
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