Erdős problem 968 — wave 7x report
Date of live-page access and computation: 2026-07-28 (UTC).
Claim labels
- [a] elementary-rigorous: proved here from definitions, algebra, or elementary inequalities.
- [b] rigorous-modulo-named-theorem: the deduction is rigorous, but invokes the named published theorem/preprint.
- [c] plausible/structural-unverified: heuristic only.
- [d] computational-only: an exact finite computation, not an asymptotic theorem.
- [S] source audit: a bibliographic or live-page observation rather than a mathematical claim.
Step 0: mandatory live-page audit
[S] Access. I fetched both the live problem page and its discussion thread through the Bright Data browser path. Direct datacenter access was not used as authority. The page says it was last edited 31 March 2026.
[S] Verbatim current statement.
> Let \(u_n=p_n/n\), where \(p_n\) is the \(n\)th prime. Does the set of \(n\) such that \(u_n [S] Gate status. Thus the mandatory skip condition did not fire. [S] Results and interpretation listed on the live page. 1. Erdős and Prachar are credited with \[
\sum_{p_n and with proving that \(\{n:u_n>u_{n+1}\}\) has positive density. 2. The site interprets Erdős's “positive density” here as positive lower density. 3. The page also records Erdős's questions whether either \(u_n \(u_n>u_{n+1}>u_{n+2}\) occurs infinitely often. [S] The two live comments, both by Terence Tao. 1. 8 Sep 2025 (explicitly marked as completely rewritten after an earlier sign error). It records the exact rearrangement \(u_n 2. 10 Sep 2025. It gives a conditional affirmative argument assuming RH and estimate (5) in Heath-Brown's pair-correlation paper. The key quoted consequence is a first-moment tail bound \[
\sum_{\substack{p_n\le x\\p_{n+1}-p_n\ge A\log x}}
(p_{n+1}-p_n)\ll x/A.
\] Together with the GPY small-gap result and conservation of the total gap length, this forces a positive proportion of moderately large gaps. The comment also says that a sufficiently uniform prime-tuples conjecture implies the answer and that this appears close to the limit of present methods. No comment claims an unconditional solution. [S] Original source. In Erdős, Some Recent Advances and Current Problems in Number Theory (1965), pp. 196–244, the discussion on printed pp. 203–204 says that he cannot prove positive upper density for gaps exceeding the average scale, then defines \(u_k=p_k/k\), records the positive-density decreasing result, and says he cannot prove the analogous increasing result. This is the same question as the live statement. [S] Erdős–Prachar. The six-page primary paper is P. Erdős and K. Prachar, Sätze und Probleme über \(p_k/k\), Abh. Math. Sem. Univ. Hamburg 25 (1961/62), 251–256. Its final page states the two lower-density questions and proves the decreasing case; its Theorem 1 contains the total-variation estimate quoted by the live page. [b] Small-gap input. Goldston–Pintz–Yıldırım, Primes in tuples IV: Density of small gaps between consecutive primes, Acta Arith. 160 (2013), 37–53, DOI 10.4064/aa160-1-3, Theorem 1, proves for every sufficiently small fixed \(\eta>0\) [b] Mean-square input. Julia Stadlmann, On the mean square gap between primes, arXiv:2212.10867, Theorem 1, states unconditionally that, for every fixed \(\varepsilon>0\), I checked the displayed theorem and proof endpoint in the 71-page primary preprint. This is still cited as a preprint in 2026 sources, so every use below is explicitly labeled “rigorous modulo Stadlmann's Theorem 1”; no peer-review claim is made. [S] Conditional source. D. R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta-function, Acta Arith. 41 (1982), 85–99, DOI 10.4064/aa-41-1-85-99 exists and its estimate (5) and Corollary 1 are the pair-correlation input cited in Tao's comment. [S] Search miss, stated narrowly. Exact-phrase searches for \(p_n/n\), positive-density large normalized prime gaps, the current mean-square record, and papers citing the above sources did not locate an unconditional positive-density solution to this exact question. The live page's March 2026 edit and September 2025 comments likewise record none. This is not a claim of an exhaustive bibliographic proof of novelty; the corollary below is presented as a verified deduction, not as a claimed new theorem of record. Put Then Consequently Both (1) and (2) are [a] elementary-rigorous. Equation (2) is also the integer-only test used by the checker. For every \(\varepsilon>0\), if Stadlmann's Theorem 1 is used as a named input, then
Primary-source literature audit
Exact reformulation
Main partial result
Theorem
Claims (3) and (4) are [b] rigorous modulo GPY Theorem 1, Stadlmann Theorem 1, and the prime number theorem. They are unconditional with respect to RH, pair correlation, and prime-tuples conjectures, but they do not give positive density.
Proof
Fix a sufficiently small constant \(\eta<1/4\) for which the local form (GPY) holds. Let
\[ \mathcal I_X=\{n:XThe prime number theorem gives, uniformly for \(n\in\mathcal I_X\),
\[ a_n=\frac{p_n}{n}=(1+o(1))\log p_n=(1+o(1))\log X. \]For all large \(X\), therefore, \(a_n\ge(3/4)\log X\). Every GPY gap
\(g_n\le\eta\log X\) has
\[ -e_n=a_n-g_n\ge(3/4-\eta)\log X\ge\tfrac12\log X. \]There are \(\gg_\eta X/\log X\) such gaps. Thus, if
\[ D_X=\sum_{\substack{n\in\mathcal I_X\\e_n<0}}(-e_n), \]then
\[ D_X\gg_\eta X. \tag{5} \]2. The signed mass is only \(o(X)\). [b]
Let \(m=\pi(X)\) and \(M=\pi(2X)\). Telescoping gives
\[ \sum_{n\in\mathcal I_X}g_n =\sum_{n=m+1}^{M}(p_{n+1}-p_n) =p_{M+1}-p_{m+1} =(1+o(1))X. \tag{6} \]Also, uniformly on this index interval, \(a_n=(1+o(1))\log X\), while
\[ M-m=(1+o(1))X/\log X. \]Hence
\[ \sum_{n\in\mathcal I_X}a_n=(1+o(1))X. \tag{7} \]Subtracting (7) from (6),
\[ \sum_{n\in\mathcal I_X}e_n=o(X). \tag{8} \]Let
\[ P_X=\sum_{\substack{n\in\mathcal I_X\\e_n>0}}e_n. \]Since \(\sum e_n=P_X-D_X\), equations (5) and (8) imply
\[ P_X\gg_\eta X. \tag{9} \]This is the precise conservation argument: the positive gaps must carry linear total excess, even though their count remains unknown.
3. The mean-square theorem prevents all excess from sitting in too few gaps. [a+b]
Let
\[ B_X=\#\{n\in\mathcal I_X:e_n>0\}. \]For \(e_n>0\), one has \(0 Combining (9) and (10), which is (3). The Cauchy step and exponent arithmetic are [a]; insertion of (MS) makes the conclusion [b]. 4. Passage to index order. [a+b] It suffices first to take \(0<\varepsilon<77/100\); the cases with larger \(\varepsilon\) follow from any stronger bound with a smaller positive exponent. For a large integer \(N\), apply (3) with \(X=p_N/2\). Every counted prime is at most \(p_N\), hence its index is at most \(N\). Since \(p_N\ge N\), This proves (4). [b] It gives a polynomial quantitative strengthening of mere infinitude: every sufficiently large dyadic prime-value interval contains at least \(X^{0.77-o(1)}\) upward steps. [a] It does not prove the requested positive density. Dividing (4) by \(N\) leaves only \(N^{-0.23-o(1)}\), which may tend to zero. No uniformity or finiteness step has been suppressed. The balancing argument already supplies \(P_X\gg X\). What is missing is a theorem preventing a fixed positive fraction of this mass from living in gaps much larger than \(\log X\). A sufficient first-moment tail lemma is If (11) held, choose a fixed \(A\) so that the tail contributes less than half of (9). The remaining positive excess is \(\gg X\), each remaining gap is at most \(A\log X\), and therefore there are \(\gg X/\log X\) positive-excess gaps. This would prove positive lower density. This implication is [a] elementary-rigorous. The sharp-order mean-square estimate would also suffice immediately in (10), giving \(B_X\gg X/\log X\). This implication is [a]; estimate (12) itself is not known unconditionally here. Stadlmann's present bound only yields which is larger than \(x\) for every fixed \(A\) once \(x\) is large. Thus it cannot establish (11). Equation (13) is [a+b] and identifies the precise \(x^{0.23+\varepsilon}\) loss. Heath-Brown's pair-correlation consequence quoted in the live comment supplies an \(O(x/A)\) version of the required tail estimate, but only under RH plus the stated pair-correlation hypothesis. That conditional route is [b], not an unconditional solution. The standalone checker is It uses no prime library or downloaded prime table. A segmented odd-only Eratosthenes sieve and a separately implemented dense odd-only Eratosthenes sieve each recompute all required primes and must produce identical result objects. A third trial-division generator checks (1), (2), telescoping, the positive/negative mass split, and the finite Cauchy inequality with exact The core event test is: Both full sieves produced the following table. Every entry is [d] computational-only but exact. | \(N\) | \(p_N\) | \(p_{N+1}\) | \(A(N)\) | \(A(N)/N\) | |---:|---:|---:|---:|---:| | 10 | 29 | 31 | 6 | 0.600000000000 | | 100 | 541 | 547 | 60 | 0.600000000000 | | 1,000 | 7,919 | 7,927 | 475 | 0.475000000000 | | 10,000 | 104,729 | 104,743 | 4,463 | 0.446300000000 | | 100,000 | 1,299,709 | 1,299,721 | 41,299 | 0.412990000000 | | 1,000,000 | 15,485,863 | 15,485,867 | 406,140 | 0.406140000000 | | 10,000,000 | 179,424,673 | 179,424,691 | 4,212,774 | 0.421277400000 | The exhaustive prefix scan gives the sharper finite assertion with equality at \(m=598768\). Thus every integer \(m\) in that entire finite range satisfies the corresponding exact lower bound. Claim (14) is [d] and says nothing about \(m>10^7\). Additional exact certificates are [d]: Reproduction command: The recorded run completed both full sieves plus exact audits in 21.54 seconds of wall time, used 21.34 user CPU seconds, and peaked at 162,220 KiB RSS. It printed: No heavier computation is proposed: extending a finite table cannot establish the uniform tail statement (11), which is the analytic obstruction. [b] The verified analytic progress is the unconditional-with-respect-to-conjectures bound \(A(N)\gg_\varepsilon N^{77/100-\varepsilon}\), modulo the named GPY and Stadlmann theorems. [d] The exact finite progress is the complete table and sharp prefix-density bound through \(10^7\). [a] Positive density still requires a tail-uniformity input such as (11), or a sharp mean-square estimate such as (12). PARTIAL: GPY small-gap density plus Stadlmann's mean-square theorem gives \(A(N)\gg_\varepsilon N^{77/100-\varepsilon}\), and two independent exact sieves verify \(A(10^7)=4,212,774\); positive density remains open because the required fixed-\(A\) large-gap tail bound is unavailable.What this does and does not settle
Exact remaining wall
Exact computation through \(n=10^7\)
runs/erdos968_wave7x_reverify.py.Fraction arithmetic.gap = p_next - p_n
goes_up = n * gap > p_n
python3 runs/erdos968_wave7x_reverify.py
exact structural audit: PASS
segmented sieve: PASS
dense sieve: PASS
independent full-table agreement: PASS
hard-coded certificate comparison: PASS
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