Erdős problem #1184 — wave 6x

Date: 2026-07-27 UTC

0. Mandatory live-page gate

I fetched https://www.erdosproblems.com/1184 and its linked discussion thread through the Bright Data browser. I did not infer the statement from the tracker metadata or use a datacenter curl response as a substitute.

The live page showed:

• status: OPEN;
• 0 claimed proofs for this problem;
• Currently working on this problem: None;
• Interested in collaborating: None;
• likes, “looks difficult”, “looks tractable”, “results could be

formalisable”, and “working on formalising”: all None;

• last page edit: 06 April 2026.

Thus the mandatory stop condition did not fire.

Verbatim current statement

Let \(f(n,k)\) count the number of \(1\leq i\leq k\) such that \(P(n+i)>k\) (where \(P(m)\) is the largest prime divisor of \(m\)). Is it true that, if \(\alpha>1\) is such that \(n=k^{\alpha+o(1)}\), then \[ > f(n,k)=(1-\rho(\alpha)+o(1))k, > \] where \(\rho\) is the Dickman function?

Verbatim listed known results

Erdős [Er76e] proved that, for every \(\alpha>1\), if \(k\) is sufficiently large and \(n>k^\alpha-k\) then \[ > f(n,k)>\left(1-\frac1\alpha+c_\alpha\right)k > \] for some constant \(c_\alpha>0\), and if \(1<\alpha<2\) and \(n\leq k^\alpha-k\) then \[ > f(n,k)<(\alpha-1+o(1))k. > \] He knew of no non-trivial bounds when \(\alpha\geq2\).

Ramachandra, Shorey, and Tijdeman [RST75b] proved that if \(n>\exp(c(\log k)^2)\) for a constant \(c>0\) then \(f(n,k)\geq k-\pi(k)\).

The one current comment

The linked thread is https://www.erdosproblems.com/forum/thread/1184. Its only comment is:

I believe the problem statement should read as \(f(n,k)=(1-\rho(\alpha)+o(1))k\) instead.

It was posted by JakeMallen at 01:02 on 05 April 2026. The revision history shows that the preceding page version had \(\rho(1/\alpha)\); the current page has incorporated the correction to \(\rho(\alpha)\). The site labels comments as unverified user content. There are no claimed proofs or worker markers in the thread.

1. Claim labels

Every mathematical assertion below is labelled:

• [a] elementary-rigorous;
• [b] rigorous modulo the explicitly named theorem(s);
• [c] plausible/structural-unverified;
• [d] computational-only.

Live-page observations and bibliographic metadata are factual audit notes, rather than mathematical claims.

2. Exact reformulation as a short-interval smooth-number problem

Write

For integers \(n>k\geq2\), every one of \(n+1,\ldots,n+k\) is either \(k\)-smooth or has largest prime factor greater than \(k\). Therefore

This is an exact identity. [a]

Consequently, the live question is exactly the all-starting-points short-interval asymptotic

It is not merely the classical long-interval Dickman theorem. [a]

3. Primary-source literature audit

3.1 Original sources listed by the page

The official journal scan of P. Erdős, Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), 271–282, exists. Page 272 states equation (3) in the equivalent form

where the preceding paragraph defines \(c_\alpha\) as the density of \(k\)-smooth integers up to \(k^\alpha\), and notes \(c_\alpha=1-\log\alpha\) for \(1\leq\alpha\leq2\). The same page states the two displayed Erdős bounds and attributes the \(k-\pi(k)\) result to Ramachandra, Shorey, and Tijdeman. This directly confirms that the current \(\rho(\alpha)\), not \(\rho(1/\alpha)\), is the intended parameter. [a]

The cited Grimm paper also exists: K. Ramachandra, T. N. Shorey, and R. Tijdeman, On Grimm's problem relating to factorisation of a block of consecutive integers. II, J. Reine Angew. Math. 288 (1976), 192–201. I use the live page's stated consequence as ground truth and do not need to strengthen it here.

3.2 A 2024 theorem not listed on the live page

Khalid Younis, Asymptotics for smooth numbers in short intervals, arXiv:2409.05761v1 (9 September 2024), is a real 30-page primary preprint. Its Theorem 1.1 says that for fixed \(17/30<\theta\leq1\), there is \(C=C(\theta)>0\) such that

uniformly for

Thus (3.1) is a relative asymptotic. The arXiv source file itself was downloaded and checked, rather than reconstructing the formula from a search snippet. [b: Younis, Theorem 1.1]

Theorem 1.3 of the same paper states, assuming the Riemann hypothesis, an analogous result for every fixed \(1/2<\theta\leq1\), in the much larger range \((\log x)^{K(\theta)}\leq y\leq2x\); its main term has an extra saddle-point factor which is \(1+o(1)\) when \(u\) is bounded. [b: RH and Younis, Theorem 1.3, together with its displayed saddle-point estimate]

The recent peer-reviewed paper of Sarvagya Jain, Existence of Smooth Numbers in Short Intervals, Q. J. Math. 77 (2026), 397–422, explicitly records Younis's \(x^{17/30+o(1)}\) all-interval asymptotic as the current asymptotic input. Jain proves stronger existence results in other ranges, not the density asymptotic (2.2), so it does not extend the result below.

The needed global estimate is classical. For example, A. Hildebrand, On the number of positive integers \(\leq x\) and free of prime factors \(>y\)90013-2), J. Number Theory 22 (1986), 289–307, gives in the present power-sized \(y\) regime

uniformly when \(\log x/\log y\) remains in a fixed compact subset of \((1,\infty)\). [b: the Dickman–de Bruijn/Hildebrand theorem]

I also searched exact strings from #1184, the expression P(n+i)>k, the title and citations of the original paper, and 2024–2026 primary papers on asymptotics for smooth numbers in all short intervals. I found no newer primary theorem improving Younis's \(17/30\) all-interval asymptotic exponent. This is a documented search miss, not a claim that no such work can exist.

4. Main result: an unconditional solved range

Proposition

For every fixed

and every integer sequence \(k\to\infty\), \(n=k^{\alpha+o(1)}\), the conclusion of Erdős problem #1184 holds:

[b: Younis, Theorem 1.1, and the classical Dickman–de Bruijn/Hildebrand theorem]

Proof

Set

[a]

Because \(\alpha<30/17\), choose a fixed number

Then

so \(x^\theta\leq h\leq x\) for all sufficiently large \(k\). Also \(y=k\leq2n=2x\). [a]

For the lower condition on \(y\), with the fixed constant \(C=C(\theta)\) from Younis,

Hence

eventually. Thus every hypothesis of Younis's Theorem 1.1 is met. [a]

Younis therefore gives

where the middle parenthesis is multiplicative. [b: Younis, Theorem 1.1]

Since \(u=\alpha+o(1)\) remains bounded, the global Dickman theorem and continuity of \(\rho\) give

[b: Dickman–de Bruijn/Hildebrand]

Combining (4.4), (4.5), and the exact complement identity (2.1) proves (4.1). [a]

The inequalities are strict: Younis requires \(\theta>17/30\), so this argument does not include \(\alpha=30/17\). [a]

Conditional extension

Assume the Riemann hypothesis. For any fixed \(1<\alpha<2\), choose \(1/2<\theta<1/\alpha\). Then \(h=k\geq n^\theta\) eventually and \(k\geq(\log n)^{K(\theta)}\). Younis's Theorem 1.3 applies; its saddle-point factor is \(1+o(1)\) for \(\log n/\log k=\alpha+o(1)\). The same Dickman and complement steps give (4.1). Thus RH implies #1184 for every fixed

[b: RH, Younis Theorem 1.3, and Dickman–de Bruijn/Hildebrand]

Again the endpoint \(\alpha=2\) is not included because the theorem requires \(\theta>1/2\). [a]

5. Exact subquadratic prime-sum identity

When

there is also the exact formula

[a]

Indeed, an integer \(m\leq n+k\) cannot have two prime factors greater than \(k\), counted with multiplicity, because their product would be at least \((k+1)^2>m\). Thus every \(m\in(n,n+k]\) counted by \(f\) has a unique representation

Conversely, every such product has largest prime factor \(p>k\). For fixed \(a\), the allowable primes are exactly

which proves (5.2). [a]

For a fixed \(1<\alpha<2\), condition (5.1) holds eventually whenever \(n=k^{\alpha+o(1)}\). On \(1\leq\alpha\leq2\), the defining differential-delay equation of the Dickman function integrates to

Hence in this range the conjectured conclusion is equivalently

[a]

Formula (5.2) is a clean reduction, but applying a prime number theorem separately to all its summands is too demanding. Near \(a\asymp n/k\), the prime interval has base point \(n/a\asymp k\) and length

Thus for \(\alpha\) near \(2\), a termwise approach asks for primes in every extremely short interval. It also loses the aggregate structure that Younis's smooth-number method exploits. [a]

6. Reproducible exact computation

The standalone checker is:

runs/erdos1184_wave6x_reverify.py

Run it with:

python3 runs/erdos1184_wave6x_reverify.py

It uses only the Python standard library. It constructs largest-prime- factor and prime-counting arrays from scratch and checks that array against independent trial division through 10,000.

First, it exhaustively verified (5.2) for every

a total of exactly 73,809 pairs \((n,k)\). [d]

Second, set \(A=\lfloor k^{3/2}\rfloor\). For each displayed \(k\), the checker examined every integer \(n\in[A,2A]\), not a sample. It computed the exact minimum and maximum of \(f(n,k)\), their first locations, and the mean of \(f(n,k)/k\). This dyadic box satisfies \(n=k^{3/2+O(1/\log k)}\). [a]

[d]

For comparison, the limiting prediction at \(\alpha=3/2\) is

[a]

The first scan obtains each rolling-window count from the largest-factor array. Independently, the checker uses a separate Eratosthenes prime list, divides every integer in the full union of the scanned windows by all prime powers at most \(k\), constructs a second indicator array, and repeats each entire scan. The two methods agree on the total, minimum, maximum, and first extremizers in every row. The complete run finished in about 3.5 seconds on this VM and printed ALL CHECKS PASSED. [d]

These finite values are still far from (6.1) and prove no asymptotic statement; they are reported only as exact reproducible data. [d]

7. What remains, precisely

Unconditionally, the argument above leaves

under RH it leaves \(\alpha\in[2,\infty)\). These are statements about the reach of the cited theorems and this deduction, not assertions that no other method can work. [b]

The exact missing analytic input for a given remaining fixed \(\alpha\) is an all-interval, not almost-all, estimate

uniformly along

For \(30/17\leq\alpha<2\), it would suffice to lower Younis's admissible all-interval exponent from \(\theta>17/30\) to some \(\theta<1/\alpha\). At \(\alpha\geq2\), one needs an asymptotic at or below the square-root scale. [a]

Almost-all short-interval theorems do not establish (7.1), because the sequence \(n=n(k)\) in #1184 is arbitrary and could, in principle, lie inside the exceptional sets for every \(k\). Existence of one smooth number is also far weaker than the required \((\rho(\alpha)+o(1))k\) count. [a]

Brute force cannot supply the uniform limiting step. As a scale estimate, a monolithic exact scan of the full dyadic range at \(\alpha=2\), \(k=10^6\), would address about \(2\cdot10^{12}\) integers. A four-byte largest-factor array alone would require about 8 TB, and a classical sieve would perform on the order of \(7\cdot10^{12}\) factor updates—roughly \(10^2\)–\(10^3\) core-hours after realistic segmented-factorization and memory overhead. Even that finite calculation would not prove (7.1). [d]

PARTIAL: Younis's 2024 all-interval theorem plus the classical Dickman theorem proves #1184 for every fixed \(1<\alpha<30/17\) unconditionally (and for \(1<\alpha<2\) under RH); the endpoints and larger \(\alpha\) remain beyond the cited machinery, with an exact subquadratic prime-sum reduction and independently verified finite scans supplied.