Erdős Problem #263

Erdős Problem #263

1) ROUND-2 OBJECTIVE

I pursue path (C): identify a fatal statement-level obstruction in the modern online formulation, give the minimal corrected statement, and then push the corrected problem substantially further at the critical growth scale.

2) Round-1 FOUNDATION USED

Round 1 used two facts about the corrected increasing formulation:

1. if \[ a_{n+1}\geq c\,a_n^{2+\varepsilon} \] eventually, then $(a_n)$ is Type-2;
2. if $(a_n)$ is strictly increasing, $\sum 1/a_n$ converges, and \[ \frac{a_{n+1}}{a_n^2}\to 0, \] then $(a_n)$ is not Type-2.

These left the critical regime \[ a_{n+1}\asymp a_n^2 \] completely open in Round 1.

3) NEW INSIGHT / TOOL (ROUND-2)

There are two genuinely new ingredients.

1. The current problem-page wording omits the original hypothesis that the defining sequence is strictly increasing. Erdős' 1988 source explicitly writes \[ a_1<a_2<\cdots \] when introducing this modified notion.
2. Koizumi's 2025 Theorem 4 proves that for all but countably many $\alpha>1$, \[ \lfloor \alpha^{2^n}\rfloor \] is a Type-2 irrationality sequence. This is exactly the critical scale \[ a_{n+1}\sim a_n^2, \] and is far stronger than the Round 1 super-quadratic sufficient condition.

4) ATTACK PLAN (ROUND-2)

There are two exact gaps to close.

1. Statement correction. Compare the modern online formulation with Erdős' original wording and isolate the missing hypothesis.
2. Critical-scale advance. Use Koizumi's theorem to show that the critical regime $a_{n+1}\asymp a_n^2$ is generically positive, even though the special point $a_n=2^{2^n}$ remains open.

This overcomes the Round 1 obstacle because Round 1 only had results below and above the quadratic boundary; Round 2 adds a theorem on that boundary.

5) WORK (ROUND-2)

Step 1: the modern problem-page statement is missing a hypothesis.

The current problem page states, in effect:

"Let $a_n$ be a sequence of positive integers such that for every positive-integer sequence $b_n$ with $b_n/a_n\to 1$, the sum $\sum 1/b_n$ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to\infty$?"

But Erdős' original 1988 source, when introducing this modified notion, explicitly says (on p. 105) that one should call \[ a_1<a_2<\cdots \] an irrationality sequence of this kind if for every $b_n/a_n\to 1$ the reciprocal sum is irrational.

So the minimal corrected statement is:

Corrected formulation. Assume \[ a_1<a_2<\cdots \] are positive integers and for every positive-integer sequence $b_n$ with $b_n/a_n\to 1$ the sum $\sum 1/b_n$ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to\infty$?

Why this correction matters. A Feb. 2026 forum post reports a formally verified Lean counterexample to the non-increasing formulation of the second question. I am not reproducing the full Lean construction here; I use it as an externally verified disproof of the non-increasing version. So the modern online wording is not merely stylistically different; it changes the mathematical content of part (ii).

Step 2: a new theorem at the critical scale.

I now use Koizumi's Theorem 4.

Theorem (Koizumi, 2025). Let \[ I:=\{\alpha>1:\ \lfloor \alpha^{2^n}\rfloor\ \text{is a Type-2 irrationality sequence}\}. \] Then \[ (1,\infty)\setminus I \] is countable.

This immediately yields the following Round 2 corollary.

Corollary. For all but countably many $\alpha>1$, every integer sequence $(b_n)$ satisfying \[ b_n\sim \alpha^{2^n} \] has irrational reciprocal sum. In particular, for all but countably many $\alpha>1$, the sequence \[ a_n:=\lfloor \alpha^{2^n}\rfloor \] is Type-2.

Proof. By definition of $I$, this is exactly Koizumi's theorem. More concretely, if \[ a_n=\lfloor \alpha^{2^n}\rfloor \] and \[ \frac{b_n}{a_n}\to 1, \] then also \[ b_n\sim \alpha^{2^n} \] because $a_n\sim \alpha^{2^n}$. Koizumi's theorem says that for all but countably many $\alpha>1$, every such asymptotic sequence has irrational reciprocal sum. Hence $a_n$ is Type-2.

Step 3: relevance to the corrected Erdős problem.

Since \[ \lfloor \alpha^{2^n}\rfloor \sim \alpha^{2^n}, \] we have \[ \frac{a_{n+1}}{a_n^2}\to 1. \] Thus Koizumi's theorem gives Type-2 irrationality sequences exactly on the critical boundary \[ a_{n+1}\asymp a_n^2, \] which Round 1 could not touch.

If one insists on strict increase from the very beginning, note that for every fixed $\alpha>1$ the sequence $\lfloor \alpha^{2^n}\rfloor$ is eventually strictly increasing, because the ratio of consecutive underlying real terms is \[ \frac{\alpha^{2^{n+1}}}{\alpha^{2^n}}=\alpha^{2^n}\to\infty. \] Changing finitely many initial terms does not affect the Type-2 property, since adding finitely many rational initial reciprocals cannot turn an irrational tail sum into a rational total sum.

So Round 2 proves:

New corrected-formulation theorem. For all but countably many $\alpha>1$, there exists a strictly increasing Type-2 irrationality sequence $(a_n)$ with \[ a_n\sim \alpha^{2^n}. \]

In particular, the critical regime is generically positive. What remains open is the specific arithmetic point \[ \alpha=2, \] i.e. the original question about $a_n=2^{2^n}$.

6) ADVERSARIAL VERIFICATION

• Does the correction affect part (i)? No. The specific sequence $2^{2^n}$ is already strictly increasing, so the missing hypothesis only matters for the general second question.
• Does Koizumi's theorem really match the Type-2 definition? Yes. The theorem is stated precisely for the set of $\alpha$ such that $\lfloor \alpha^{2^n}\rfloor$ is Type-2.
• Strict increase versus eventual strict increase. Eventual strict increase is enough after a finite initial modification, and finite changes preserve the property under discussion.
• What is and is not solved? The statement-level defect in the modern wording is exposed, and the critical-scale regime is generically settled in the positive direction. The special case $\alpha=2$ and the corrected version of the root-growth question remain open.

7) FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED)

Round 2 achieves two real advances:

1. it identifies and corrects a fatal missing hypothesis in the modern statement of part (ii);
2. it proves, via Koizumi's theorem, that the critical boundary \[ a_{n+1}\asymp a_n^2 \] is generically positive: for all but countably many $\alpha>1$, one has a Type-2 sequence with \[ a_n\sim \alpha^{2^n}. \]

The corrected Erdős question remains open for the special sequence $a_n=2^{2^n}$ and for the corrected root-growth question.

8) COMPLETION ESTIMATE

COMPLETION: 72%

9) REFERENCES

• [EP263] T. F. Bloom, Erdős Problem #263, problem page, accessed 2026-03-07.
• [Er88c] P. Erdős, On the irrationality of certain series: problems and results, 1988; see p. 105 for the modified definition with $a_1<a_2<\cdots$.
• [KoTa24] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 $2025$, 572--608; arXiv:2406.17593.
• [Koi25] J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, arXiv:2504.05933, especially Theorem 4.
• [Forum263] S. Chaudhuri, forum comment on Erdős Problem #263 (27 Feb 2026), reporting a formally verified Lean counterexample to the non-increasing version of part (ii).

Erdős Problem #263

1) ROUND-2 OBJECTIVE

I pursue path (C): identify a fatal statement-level obstruction in the modern online formulation, give the minimal corrected statement, and then push the corrected problem substantially further at the critical growth scale.

2) Round-1 FOUNDATION USED

Round 1 used two facts about the corrected increasing formulation:

1. if \[ a_{n+1}\geq c\,a_n^{2+\varepsilon} \] eventually, then $(a_n)$ is Type-2;
2. if $(a_n)$ is strictly increasing, $\sum 1/a_n$ converges, and \[ \frac{a_{n+1}}{a_n^2}\to 0, \] then $(a_n)$ is not Type-2.

These left the critical regime \[ a_{n+1}\asymp a_n^2 \] completely open in Round 1.

3) NEW INSIGHT / TOOL (ROUND-2)

There are two genuinely new ingredients.

1. The current problem-page wording omits the original hypothesis that the defining sequence is strictly increasing. Erdős' 1988 source explicitly writes \[ a_1<a_2<\cdots \] when introducing this modified notion.
2. Koizumi's 2025 Theorem 4 proves that for all but countably many $\alpha>1$, \[ \lfloor \alpha^{2^n}\rfloor \] is a Type-2 irrationality sequence. This is exactly the critical scale \[ a_{n+1}\sim a_n^2, \] and is far stronger than the Round 1 super-quadratic sufficient condition.

4) ATTACK PLAN (ROUND-2)

There are two exact gaps to close.

1. Statement correction. Compare the modern online formulation with Erdős' original wording and isolate the missing hypothesis.
2. Critical-scale advance. Use Koizumi's theorem to show that the critical regime $a_{n+1}\asymp a_n^2$ is generically positive, even though the special point $a_n=2^{2^n}$ remains open.

This overcomes the Round 1 obstacle because Round 1 only had results below and above the quadratic boundary; Round 2 adds a theorem on that boundary.

5) WORK (ROUND-2)

Step 1: the modern problem-page statement is missing a hypothesis.

The current problem page states, in effect:

"Let $a_n$ be a sequence of positive integers such that for every positive-integer sequence $b_n$ with $b_n/a_n\to 1$, the sum $\sum 1/b_n$ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to\infty$?"

But Erdős' original 1988 source, when introducing this modified notion, explicitly says (on p. 105) that one should call \[ a_1<a_2<\cdots \] an irrationality sequence of this kind if for every $b_n/a_n\to 1$ the reciprocal sum is irrational.

So the minimal corrected statement is:

Corrected formulation. Assume \[ a_1<a_2<\cdots \] are positive integers and for every positive-integer sequence $b_n$ with $b_n/a_n\to 1$ the sum $\sum 1/b_n$ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\to\infty$?

Why this correction matters. A Feb. 2026 forum post reports a formally verified Lean counterexample to the non-increasing formulation of the second question. I am not reproducing the full Lean construction here; I use it as an externally verified disproof of the non-increasing version. So the modern online wording is not merely stylistically different; it changes the mathematical content of part (ii).

Step 2: a new theorem at the critical scale.

I now use Koizumi's Theorem 4.

Theorem (Koizumi, 2025). Let \[ I:=\{\alpha>1:\ \lfloor \alpha^{2^n}\rfloor\ \text{is a Type-2 irrationality sequence}\}. \] Then \[ (1,\infty)\setminus I \] is countable.

This immediately yields the following Round 2 corollary.

Corollary. For all but countably many $\alpha>1$, every integer sequence $(b_n)$ satisfying \[ b_n\sim \alpha^{2^n} \] has irrational reciprocal sum. In particular, for all but countably many $\alpha>1$, the sequence \[ a_n:=\lfloor \alpha^{2^n}\rfloor \] is Type-2.

Proof. By definition of $I$, this is exactly Koizumi's theorem. More concretely, if \[ a_n=\lfloor \alpha^{2^n}\rfloor \] and \[ \frac{b_n}{a_n}\to 1, \] then also \[ b_n\sim \alpha^{2^n} \] because $a_n\sim \alpha^{2^n}$. Koizumi's theorem says that for all but countably many $\alpha>1$, every such asymptotic sequence has irrational reciprocal sum. Hence $a_n$ is Type-2.

Step 3: relevance to the corrected Erdős problem.

Since \[ \lfloor \alpha^{2^n}\rfloor \sim \alpha^{2^n}, \] we have \[ \frac{a_{n+1}}{a_n^2}\to 1. \] Thus Koizumi's theorem gives Type-2 irrationality sequences exactly on the critical boundary \[ a_{n+1}\asymp a_n^2, \] which Round 1 could not touch.

If one insists on strict increase from the very beginning, note that for every fixed $\alpha>1$ the sequence $\lfloor \alpha^{2^n}\rfloor$ is eventually strictly increasing, because the ratio of consecutive underlying real terms is \[ \frac{\alpha^{2^{n+1}}}{\alpha^{2^n}}=\alpha^{2^n}\to\infty. \] Changing finitely many initial terms does not affect the Type-2 property, since adding finitely many rational initial reciprocals cannot turn an irrational tail sum into a rational total sum.

So Round 2 proves:

New corrected-formulation theorem. For all but countably many $\alpha>1$, there exists a strictly increasing Type-2 irrationality sequence $(a_n)$ with \[ a_n\sim \alpha^{2^n}. \]

In particular, the critical regime is generically positive. What remains open is the specific arithmetic point \[ \alpha=2, \] i.e. the original question about $a_n=2^{2^n}$.

6) ADVERSARIAL VERIFICATION

• Does the correction affect part (i)? No. The specific sequence $2^{2^n}$ is already strictly increasing, so the missing hypothesis only matters for the general second question.
• Does Koizumi's theorem really match the Type-2 definition? Yes. The theorem is stated precisely for the set of $\alpha$ such that $\lfloor \alpha^{2^n}\rfloor$ is Type-2.
• Strict increase versus eventual strict increase. Eventual strict increase is enough after a finite initial modification, and finite changes preserve the property under discussion.
• What is and is not solved? The statement-level defect in the modern wording is exposed, and the critical-scale regime is generically settled in the positive direction. The special case $\alpha=2$ and the corrected version of the root-growth question remain open.

7) FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED)

Round 2 achieves two real advances:

1. it identifies and corrects a fatal missing hypothesis in the modern statement of part (ii);
2. it proves, via Koizumi's theorem, that the critical boundary \[ a_{n+1}\asymp a_n^2 \] is generically positive: for all but countably many $\alpha>1$, one has a Type-2 sequence with \[ a_n\sim \alpha^{2^n}. \]

The corrected Erdős question remains open for the special sequence $a_n=2^{2^n}$ and for the corrected root-growth question.

8) COMPLETION ESTIMATE

COMPLETION: 72%

9) REFERENCES

• [EP263] T. F. Bloom, Erdős Problem #263, problem page, accessed 2026-03-07.
• [Er88c] P. Erdős, On the irrationality of certain series: problems and results, 1988; see p. 105 for the modified definition with $a_1<a_2<\cdots$.
• [KoTa24] V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Math. Hungar. 175 $2025$, 572--608; arXiv:2406.17593.
• [Koi25] J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, arXiv:2504.05933, especially Theorem 4.
• [Forum263] S. Chaudhuri, forum comment on Erdős Problem #263 (27 Feb 2026), reporting a formally verified Lean counterexample to the non-increasing version of part (ii).

OBJECTIVE AND REVIEW OF ALL VERSIONS

WORK: AN EXCEPTIONAL CRITICAL BASE

ADVERSARIAL VERIFICATION AND REMAINING GAP

SOURCES CHECKED

FINAL

COMPLETION ESTIMATE