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Math-728-A2 · revision r1 · on Math-728

gpt pro 5.2 attempt on Math-728

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem #244
  2. 1) FORMAL RESTATEMENT
  3. 2) QUICK LITERATURE/CONTEXT CHECK
  4. 3) ATTACK PLAN
  5. 4) WORK
  6. 4.1 A trivial but rigorous lower bound
  7. 4.2 Small computational sanity checks (not proofs)
  8. 5) VERIFICATION
  9. 6) FINAL
  10. 7) COMPLETION ESTIMATE

Erdős Problem #244

1) FORMAL RESTATEMENT

Fix a real number $C>1$. Define the "geometric-floor" sequence \[ t_k \coloneqq \lfloor C^k\rfloor\qquad (k\in\mathbb{Z}_{\ge 0}), \] and the set \[ \mathcal{S}_C \coloneqq \{\, p + t_k : p\in\mathcal{P},\ k\in\mathbb{Z}_{\ge 0}\,\}\subseteq\mathbb{N}, \] where $\mathcal{P}$ is the set of primes and $\lfloor\cdot\rfloor$ is the floor function.

For $x\ge 1$ write \[ \mathcal{S}_C(x) \coloneqq \#\left(\mathcal{S}_C\cap\{1,2,\dots,\lfloor x\rfloor\}\right). \] The (lower) asymptotic density of $\mathcal{S}_C$ is \[ \underline{d}(\mathcal{S}_C) \coloneqq \liminf_{x\to\infty} \frac{\mathcal{S}_C(x)}{x}. \]

Question (Romanoff--Kalmár type). Is $\underline{d}(\mathcal{S}_C)>0$ for every real $C>1$?

Remark on ambiguity. The original phrasing "has density $>0$" could mean (i) the natural density exists and is positive, or (ii) the lower asymptotic density is positive. The currently recorded results (Romanoff for integer $C$; Ding for almost all real $C$) are naturally stated in terms of positive lower asymptotic density, so I interpret the question as above.

2) QUICK LITERATURE/CONTEXT CHECK

  • For integer $C=a\ge 2$, the set $\{p+a^k\}$ has positive lower asymptotic density by Romanoff (often called Romanov's theorem) [Romanoff1934]. See also the modern summary "Romanov's theorem" [WikipediaRomanov].
  • The Erdős Problems database page for #244 records: Romanoff's theorem for integer $C$, and that Y. Ding $2025$ proved positivity of the lower density for almost all $C>1$ (in the sense of Lebesgue measure) [ErdosProblems244,Ding2025].
  • As of the last edit recorded on the Erdős Problems site (Oct 28, 2025), the statement "for every $C>1$" remains open [ErdosProblems244].

3) ATTACK PLAN

A classical approach for $C\in\mathbb{Z}_{\ge 2}$ (Romanoff) is:
  1. Let $r(n)$ count representations $n=p+a^k$.
  2. Use Cauchy--Schwarz: $\#\{n\le x:r(n)>0\}\ge \frac{(\sum_{n\le x} r(n))^2}{\sum_{n\le x} r(n)^2}$.
  3. Estimate $\sum r(n)$ from below and bound $\sum r(n)^2$ from above using sieve bounds for prime pairs with fixed differences $a^{k_2}-a^{k_1}$.

For general real $C>1$, one would need to control differences \[ \lfloor C^{k_2}\rfloor-\lfloor C^{k_1}\rfloor \] uniformly in $k_1,k_2$ and prevent exceptional $C$ for which these differences have "too many" small prime factors (making sieve upper bounds too large) or create excessive overlap.

A disproof strategy (if false) would seek a specific $C$ where the set of shifts $\{\lfloor C^k\rfloor\}$ forces strong congruence restrictions so that $\mathcal{S}_C$ misses a positive proportion of residue classes for infinitely many moduli; no such explicit $C$ is currently known to me.

4) WORK

4.1 A trivial but rigorous lower bound

Even without Romanoff/Ding, one always has a nontrivial growth lower bound (though it only gives density $0$):
Lemma (Uniform lower bound from the $k=0$ shift): For every $C>1$ and every $x\ge 2$, we have \[ \mathcal{S}_C(x)\ge \pi(x-1), \] where $\pi$ is the prime-counting function.
Proof: Since $t_0=\lfloor C^0\rfloor=1$, every prime $p\le x-1$ produces an element $p+1\le x$ in $\mathcal{S}_C$. Distinct primes give distinct values, hence $\mathcal{S}_C(x)\ge \pi(x-1)$. ∎

Consequently $\mathcal{S}_C(x)/x \ge \pi(x-1)/x \sim 1/\log x$ as $x\to\infty$, so this does not approach a positive constant.

4.2 Small computational sanity checks (not proofs)

I computed the empirical density \[ \frac{\mathcal{S}_C(X)}{X} \] for a few sample values of $C$ and cutoffs $X\in\{2\cdot 10^5,10^6\}$. The computation used:
  • all distinct shifts $t_k=\lfloor C^k\rfloor\le X$;
  • all primes $p\le X$;
  • a boolean array to mark $p+t_k\le X$.
$C$$\mathcal{S}_C(2\cdot 10^5)/(2\cdot 10^5)$$\mathcal{S}_C(10^6)/(10^6)$
$2$$0.554360$$0.538970$
$1.5$$0.883265$$0.899666$
$\sqrt{2}$$0.856375$$0.864454$
$e$$0.672595$$0.665399$
$1.1$$0.999945$(not computed at $10^6$)

These values are consistent with the intuition that as $C\downarrow 1$, the shifts $\lfloor C^k\rfloor$ take many small values and the union of translates of the primes covers most integers up to moderate $X$. For larger $C$ there are fewer shifts and the observed density drops.

5) VERIFICATION

  • The proof of the lemma is straightforward and uses only that $t_0=1$.
  • The computations were double-checked for off-by-one issues by ensuring (i) primes were generated up to $X$, (ii) only shifts $\le X$ were used, and (iii) only sums $p+t_k\le X$ were marked.
  • The computational results are included only as heuristic evidence; they do not address the limiting density as $X\to\infty$.

6) FINAL

UNRESOLVED.
  1. Strongest fully proved partial result obtained here: For every $C>1$, the counting function satisfies $\mathcal{S}_C(x)\ge \pi(x-1)$ for all $x\ge 2$ (Lemma above), hence $\mathcal{S}_C(x)\gg x/\log x$.
  2. First gap preventing a full solution: One needs a uniform lower bound $\mathcal{S}_C(x)\ge \delta x$ for all large $x$ with some $\delta=\delta(C)>0$. Achieving this requires controlling overlaps between the shifted prime sets $\{p+\lfloor C^k\rfloor\}$ across different $k$, typically via second-moment/sieve estimates. I did not obtain such an overlap bound for an arbitrary fixed real $C>1$.
  3. Most promising next move: Attempt to adapt Romanoff's Cauchy--Schwarz/second-moment method to the shift sequence $\lfloor C^k\rfloor$ for an arbitrary fixed $C$. The key technical step is bounding the number of solutions to $p_1-p_2=\lfloor C^{k_2}\rfloor-\lfloor C^{k_1}\rfloor$ with $p_1,p_2$ prime using sieve bounds, and then summing these bounds over $(k_1,k_2)$.
  4. Smallest plausible counterexample (if false): A candidate would be a special constant $C$ for which the differences $\lfloor C^{k_2}\rfloor-\lfloor C^{k_1}\rfloor$ are unusually "smooth" (have many small prime factors) for many pairs $(k_1,k_2)$, inflating sieve upper bounds and potentially allowing the second-moment argument to fail. I do not know any explicit $C$ with this property.

7) COMPLETION ESTIMATE

Estimated likelihood of completion with additional work: Low ($\approx 10\%$). The known results already cover integer $C$ and almost-every $C$, suggesting that the main obstacle is controlling a thin exceptional set of $C$.