Erdős problem #962 (wave 7x)

Access date: 2026-07-28 UTC.

Claim labels

• (a) elementary-rigorous: proved in this report from definitions.
• (b) rigorous-modulo-named-theorem: the elementary deduction is complete, but a specifically named published theorem is used.
• (c) plausible/structural-unverified: heuristic or a reduction with an unproved input.
• (d) computational-only: exhaustively checked in the stated finite range, with reproducible code, but not promoted to an asymptotic theorem.

Result

The main question remains open. The useful new observation is an apparently

unrecorded connection with the Grimm function. It gives the following genuine

improvement to the upper bound recorded on the live page.

> Corollary (b). For all sufficiently large \(n\),

> \[ > k(n) \]

This follows from Theorem 1(i) of Laishram and Murty, *Grimm's Conjecture

and Smooth Numbers*, Michigan Math. J. 61 (2012), 151--160,

DOI 10.1307/mmj/1331222852.

They prove \(g(m)

\(0.45<\alpha<0.46\), where \(g(m)\) is the longest prefix

\(m+1,\ldots,m+r\) admitting distinct representative prime divisors.

Every block in problem #962 automatically has such distinct representatives.

The full two-line reduction is proved below.

This settles the fixed-power saving which Erdős called a “ridiculously weak

result” in 1976, but it does not approach

\(\log k(n)\leq(\log n)^{1/2+o(1)}\). The new upper bound only says

\(\log k(n)\leq 0.46\log n\).

The finite checker also independently certifies

Step 0: authoritative live page

The live page was fetched through the Bright Data browser path, including the

linked discussion thread, rather than by datacenter curl.

Verbatim statement

> Let \(k(n)\) be the maximal \(k\) such that there exists \(m\leq n\) such that each of the integers

> \[ > m+1,\ldots,m+k > \]

> are divisible by at least one prime \(>k\). Estimate \(k(n)\) - in particular, is it true that

> \[ > \log k(n) \leq (\log n)^{1/2+o(1)}? > \]

This is copied verbatim from the page's “View the LaTeX source” rendering.

Status, results, comments, and collision markers

The page displayed OPEN, last edited 03 April 2026. It displayed **0

claimed proofs** and the following marker values:

| Marker | Live value |

|---|---|

| Interested in collaborating | None |

| Currently working on this problem | None |

| I am working on formalising the results | None |

Thus the mandatory stop condition did not trigger.

The page records the following mathematical results.

1. (b) Erdős [Er65] wrote that

\[ \log k(n)\geq(\log n)^{1/2-o(1)} \]

and conjectured \(k(n)=o(n^\epsilon)\), but had no nontrivial upper

bound.

2. (b) Erdős [Er76e] obtained

\[ \log k(n)\gg\sqrt{\log n\log\log n} \]

and reported

\[ k(n)\leq n^{1/2}\exp\!\bigl(-(\log n)^c\bigr) \]

for some \(c>0\), while saying he could not prove

\(k(n)\leq n^{1/2-c}\) for a fixed \(c>0\).

3. (b) Tao's comment, using the prime number theorem, proves

\(k(n)\leq(1+o(1))n^{1/2}\): if

\(k>(1+\epsilon)\sqrt n\), a prime

\(\sqrt n

of length \(k\), and that multiple has no prime factor exceeding \(k\).

4. (b) Tang's note proves

\[ \log k(n)\geq \left(\frac1{\sqrt2}-o(1)\right) \sqrt{\log n\log\log n}. \]

There were five comments, all read before beginning the mathematics:

• Terence Tao, 12 October 2025: the \( (1+o(1))\sqrt n\) upper-bound

argument just summarized.

• Terence Tao, 28 October 2025: “Some initial numerics,” linking

GitHub issue #145.

That issue computes the least starting point for 400 values of \(k\).

• Quanyu Tang, 28 December 2025: announces and links the improved lower

bound above.

• Thomas Bloom, 03 April 2026: records the rediscovered 1976 Erdős bounds.
• Darin Dimitroff (deezel), 02 May 2026: proposes a fourth-moment

route which, if two level-of-distribution/bilinear estimates labelled

L5 and L6 held, would give \(k(n)\ll n^{1/e+o(1)}\). (c) The

commenter explicitly disclaims a theorem; L5 and L6 are unproved.

The numerical normality, kurtosis, and variance observations in that

comment are (d) and are not used here.

Primary-source audit and literature search

The following sources were opened and checked directly.

| Source | What was verified |

|---|---|

| Erdős, Extremal problems in number theory (1965), p. 183, Problem 5 | The original definition, the lower bound \(k(n)>\exp((\log n)^{1/2-\epsilon})\), the \(o(n^\epsilon)\) conjecture, and the statement that no nontrivial upper bound was known. |

| Erdős, Problems and results on consecutive integers (1976), p. 273 | In the inverse notation \(n_k\), the displayed bounds \(n_kk^2\exp((\log k)^c)\), the conjectured \(n_k>\exp((\log k)^{2-\epsilon})\), and the “ridiculously weak” sentence. |

| Tang, An Improved Lower Bound on Erdős Problem #962 | The note exists, is two pages, and Theorem 1.2 has the lower-bound constant \(1/\sqrt2\). |

| Laishram--Murty, Grimm's Conjecture and Smooth Numbers | The definition of \(g(n)\), Theorem 1(i), and the sentence that \(0.45<\alpha<0.46\) is permissible. |

The source-audit mode of the checker downloads these exact PDFs and verifies

their SHA-256 hashes, page counts, and extractable theorem phrases. The 1976

PDF is an image-only scan, so its page 273 was additionally inspected visually.

The verified hashes are:

Er65   b2dc8978ca143027e8d5940211a71aaf69cde9f2d771499dad881d9fb27878ea
Er76e  5c7d95ca79a407de8b1f2f5ee01e7ad6acdf304cfc587752e213e3bdf57f04fb
LM12   01724c88d7ec1c40ee81745da7a19bc95f180424a4f2aacca37993cbb5d6e8ef
Tang   3db0419ffc8d8c1bb742bb4551e53c8a93207d5b6a60879ed60fd777a6b97327

Targeted searches for the exact problem text, “Erdős Problem 962” together

with “Grimm,” the Grimm-function notation \(g(n)\), and all-interval smooth

number results found no source that already states the reduction below and no

primary source solving the main \(1/2+o(1)\) exponent question. This is a

search report, not a claim of bibliographic completeness.

The most recent directly relevant paper found was Sarvagya Jain,

Existence of Smooth Numbers in Short Intervals,

Q. J. Math. 77 (2026), 397--422. (b) Its all-interval theorem requires

an interval length at least

which is longer than \(\sqrt x\), and hence does not improve the present

worst-case exponent. Its much shorter “almost all intervals” result cannot

exclude the single exceptional interval needed to define \(k(n)\).

The Grimm reduction

For \(m>1\), Laishram and Murty say that \((m,r)\) has a *prime

representation* when there are distinct primes

and define \(g(m)\) as the largest such \(r\).

Lemma

(a) If \(m+1,\ldots,m+k\) is admissible for problem #962, then

Proof. For each \(1\leq i\leq k\), choose a prime \(q_i>k\) dividing

\(m+i\). If \(q_i=q_j=q\) for \(i \[ q\mid(m+j)-(m+i)=j-i. \]

But \(0

and give a prime representation of length \(k\). \(\square\)

Fixed power saving

(b) Laishram--Murty Theorem 1(i) says that there is a permissible

\(\alpha\in(0.45,0.46)\) and an \(m_0\) such that

runs/erdos962_wave7x_reverify.py

python3 runs/erdos962_wave7x_reverify.py

python3 runs/erdos962_wave7x_reverify.py --check-sources

def first_starts(lpf, k_max):
    starts = [0] * (k_max + 1)
    for k in range(1, k_max + 1):
        run = 0
        for value in range(1, len(lpf)):
            if lpf[value] > k:
                run += 1
                if run == k:
                    starts[k] = value - k + 1
                    break
            else:
                run = 0
        if starts[k] == 0:
            raise RuntimeError("sieve limit too small")
    return starts

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