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ERDőS #454 · PARTIAL

Erdős problem 454 — wave 5w

Accessed 2026-07-26 UTC. The labels used below are:

asymptotic theorem.

0. Mandatory live-page gate

I fetched the live page and its discussion

thread through the configured Bright Data Chromium path. I did this before

any mathematical work. The live LaTeX endpoint gives the following statement

(verbatim words and mathematics; only display whitespace has been normalized):

> Let

> \[ > f(n) = \min_{i \]

> where \(p_k\) is the \(k\)th prime. Is it true that

> \[ > \limsup_n (f(n)-2p_n)=\infty? > \]

The cited original paper writes the range explicitly as \(0

that intended range throughout. Allowing \(i=0\) would put \(2p_n\) in the

minimum and contradict both the question and its listed known result.

(d: live-page observation) The gate was clear:

The sole comment, by Terence Tao on 11 August 2025, says that the expected

important indices are extreme points of the convex hull of

\(\{(n,p_n)\}\), but that this set should be so sparse that even the prime

tuples conjecture gives little information near those indices. The site

labels comments as unverified user content, so I treat this as (c) rather

than as a theorem.

There was therefore no claimed proof and no current worker, so I proceeded.

1. Primary-source literature audit

I checked the actual documents, not just search-result summaries.

1. Erdős–Graham, 1980. Page 90 of

[*Old and New Problems and Results in Combinatorial Number

Theory*](https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf)

states this question and immediately says that Pomerance proved the

limsup is at least \(2\). I inspected the scan of page 90 directly.

(b)

2. Pomerance, 1979. Section 5 of

[“The Prime Number Graph,” Math. Comp. 33 (1979),

399–408](https://doi.org/10.1090/S0025-5718-1979-0514836-7)

defines

\[ A(n)=\min_{0

and conjectures that \(A(n)-2p_n\) is arbitrarily large. Theorem 2.1

and its corollary give infinitely many \(n\) with \(A(n)>2p_n\).

Pomerance also reports E. R. Canfield's search: for \(n\leq1000\), the

largest value was \(24\), at \(n=985\), with \(p_{985}=7759\).

The checker below independently recovers this row. **(b), with the

finite row independently (d)**

3. McNew, 2018. Section 5, equation (23), of

[“The Convex Hull of the Prime Number

Graph”](https://www.nathanmcnew.com/Convex.pdf),

DOI

10.1007/978-3-319-92777-0_7,

uses exactly

\[ M_n=\min_{1\leq i

It presents a histogram for \(n<1.6\times10^8\) and says that the data

suggest \(M_n\) is arbitrarily large. It does not prove this. The same

chapter proves results about convex-prime counts and gaps, and explicitly

distinguishes convex primes from the much more numerous midpoint-convex

primes \(M_n>0\). **(b) for what the paper proves; (c) for its

unboundedness inference**

4. I verified that Tutaj's

arXiv:1408.3609 exists and concerns

convex-hull extreme primes, but it does not resolve the minimum \(M_n\).

I also checked the full text of the recent preprint

[Kominers–Mrazović–Pomerance–Solé,

“Lines in the Prime Number Graph,” arXiv:2605.22752v3

(1 June 2026)](https://arxiv.org/abs/2605.22752). Its functions are the

minimum number of lines covering the first \(n\) prime points and the

maximum number on one line; it does not address \(M_n\).

Exact-expression and title searches found no later primary source claiming a

proof of problem 454. This is an honest search miss, not a proof that no such

paper exists. The strongest directly relevant computation I found remains

McNew's \(n<1.6\times10^8\) histogram, so the finite range below is not claimed

as a new range record.

2. Exact paired-gap formulation

Put

\[ M_n=f(n)-2p_n,\qquad g_j=p_{j+1}-p_j. \]

For \(1\leq i \[ S_n(i)=p_{n+i}+p_{n-i}-2p_n. \]

Telescoping the two sides of \(p_n\) gives

\[ \boxed{\quad S_n(i)=\sum_{r=1}^{i}\bigl(g_{n+r-1}-g_{n-r}\bigr),\qquad M_n=\min_{1\leq iThis identity is (a) and the checker tests it by two different recurrences

1,999,000 times.

Equivalently, if

\[ R_n(i)=\frac{p_{n+i}-p_n}{i},\qquad L_n(i)=\frac{p_n-p_{n-i}}{i}, \]

then

\[ S_n(i)=i\bigl(R_n(i)-L_n(i)\bigr). \tag{2} \]

Thus problem 454 asks whether the minimum vertical clearance over every

symmetric chord can be arbitrarily large. A lower-convex-hull vertex only

gives positivity of these clearances; it supplies no growing lower bound.

This is why results counting or spacing convex primes do not by themselves

settle the question. (a)

In particular,

\[ M_n\leq S_n(1)=g_n-g_{n-1}. \tag{3} \]

An unbounded positive jump between two adjacent gaps is necessary, but (1)

shows why it is far from sufficient: every longer paired partial sum must

remain above the same level. (a)

3. A PNT-error tail reduction

The standard prime-number-theorem error term removes all sufficiently long

radii from the real difficulty.

Proposition

(b: rigorous modulo the standard Vinogradov–Korobov PNT error term) There

is an absolute \(c>0\) such that, for every fixed real \(B\), all sufficiently

large \(n\) satisfy

\[ S_n(i)>B \quad\text{whenever}\quad n\exp\!\left( -c\frac{(\log n)^{3/5}}{(\log\log n)^{1/5}} \right)\leq iProof

Let \(F=\operatorname{li}^{-1}\) and put

\[ \Lambda(x)=\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}. \]

Inverting

\[ \pi(x)=\operatorname{li}(x) +O\!\left(xe^{-c_0\Lambda(x)}\right) \]

(logarithmic factors can be absorbed by reducing \(c_0\)) gives

\[ p_m=F(m)+O\!\left(me^{-c_1\Lambda(m)}\right). \tag{5} \]

This is also the form used in McNew's equation (15).

Implicit differentiation of \(\operatorname{li}(F(x))=x\) gives

\[ F'(x)=\log F(x),\qquad F''(x)=\frac{\log F(x)}{F(x)}\asymp\frac1x. \tag{6} \]

For \(i\leq n/2\), Taylor's formula in integral form and (6) therefore give

\[ F(n+i)+F(n-i)-2F(n)\gg\frac{i^2}{n}. \tag{7} \]

The three errors from (5) are

\[ O\!\left(ne^{-c_2\Lambda(n)}\right) \]

uniformly in this range, after reducing \(c_2\). Choose \(c

If \(i\geq ne^{-c\Lambda(n)}\), the right side of (7) is at least

\[ ne^{-2c\Lambda(n)}, \]

which dominates the error in (5) and tends to infinity. This proves (4)

for \(i\leq n/2\).

For \(i\geq n/2\), convexity makes

\[ F(n+i)+F(n-i)-2F(n) \]

at least its value at \(i=n/2\), which is \(\gg n\); the error in (5) is

\(o(n)\), uniformly after separating the finitely many possible small

values of \(n-i\). This completes the proof. \(\square\)

Consequently, problem 454 is equivalent to the following growing-window

statement: for every fixed \(B\), there are arbitrarily large \(n\) for which

\[ \sum_{r=1}^{i}(g_{n+r-1}-g_{n-r})>B \quad\text{for every}\quad 1\leq i< n\exp\!\left( -c\frac{(\log n)^{3/5}}{(\log\log n)^{1/5}} \right). \tag{8} \]

The implication from (8) uses (4); the reverse implication is immediate.

The constant may be reduced without changing the equivalence. (b)

Under RH, the usual

\(\pi(x)-\operatorname{li}(x)=O(x^{1/2}\log x)\) instead gives

\[ p_m=F(m)+O(m^{1/2}\log^{5/2}m). \]

The same calculation makes the tail automatic for

\[ i\geq Cn^{3/4}\log^{5/4}n \tag{9} \]

with a sufficiently large constant \(C\). Even RH therefore leaves a

window whose length grows like \(n^{3/4+o(1)}\). (b)

4. Exact prefix-record certificate through \(10^7\)

For a prefix, let

\[ H_{n-1}=\max_{2\leq mThe following short-circuit is exact:

for n in range(2, N + 1):
    for i in range(1, n):
        value = p[n+i] + p[n-i] - 2*p[n]
        if value <= H:
            save i as a non-record witness
            break
    else:
        # Every radius was scanned.
        M_n = the minimum value seen
        append (n, p_n, M_n, first_argmin) to the record table
        H = M_n

(a) Inductively, one radius with \(S_n(i)\leq H_{n-1}\) proves that \(n\)

is not a new record. If no such radius exists, every \(1\leq i

checked, so the stored minimum is exact and is a strict new record. A

certificate therefore consists of one radius for every non-record index and

a complete radius scan for every record index.

The complete standard-library checker is

erdos454_wave5w_reverify.py. It performs the

search and then audits the resulting certificate in a separate pass. Its

prime data are also independently regenerated with a structurally different

segmented sieve and compared by a digest; the first 2,000 primes are checked

again by trial division.

Run:

python runs/erdos454_wave5w_reverify.py

The complete strict prefix-record table is:

| \(n\) | \(p_n\) | \(M_n\) | first minimizing \(i\) |

|---:|---:|---:|---:|

| 2 | 3 | 1 | 1 |

| 4 | 7 | 2 | 1 |

| 21 | 73 | 4 | 1 |

| 30 | 113 | 10 | 1 |

| 189 | 1,129 | 12 | 2 |

| 217 | 1,327 | 18 | 40 |

| 985 | 7,759 | 24 | 20 |

| 1,847 | 15,823 | 26 | 10 |

| 4,612 | 44,293 | 32 | 115 |

| 9,834 | 102,701 | 38 | 1 |

| 14,357 | 155,921 | 58 | 1 |

| 63,536 | 794,249 | 68 | 1 |

| 189,689 | 2,597,117 | 70 | 1 |

| 266,856 | 3,751,919 | 72 | 3 |

| 298,595 | 4,234,537 | 78 | 1 |

| 316,504 | 4,508,341 | 82 | 1 |

| 381,415 | 5,509,453 | 88 | 2 |

| 733,588 | 11,113,933 | 90 | 6 |

| 765,401 | 11,630,503 | 118 | 1 |

| 2,886,673 | 47,973,257 | 148 | 1 |

Thus the exact finite conclusion is

\[ \boxed{\max_{2\leq n\leq10^7}M_n=148,} \]

first attained at \(n=2,886,673\). (d)

At that index the local triple is

\[ p_{n-1}=47,973,241,\quad p_n=47,973,257,\quad p_{n+1}=47,973,421. \]

The adjacent gaps are \(16\) and \(164\), so \(S_n(1)=148\); the full record

scan proves \(S_n(i)\geq148\) for every \(1\leq i \[ f(2,886,673)=95,946,662. \tag{10} \]

This is an explicit high finite example, not evidence sufficient for a

limsup theorem. (d)

The successful full run reported:

FROM-SCRATCH VERIFICATION PASSED
index range: 2 <= n <= 10,000,000
prime sequence: first 19,999,999 primes, last = 373,587,839
prime SHA-256 (u64le):
  b771acf3d077798247cabe8bd5fb044bb7e94cc1b24f2fa7a150f65e79244456
complete scan: 16,016,797 radius tests; largest single scan 2,886,672
certificate audit: 9,999,979 non-record witnesses;
                   5,934,335 record radii
direct prefix audit: n <= 2,000;
                     1,999,000 direct/gap identity checks
trial-division prime prefix: 2,000 primes
independent segmented prime audit: full digest matched
champion local triple: 47973241, 47973257, 47973421;
                       gap difference = 148
maximum resident set size: 339,020 KiB
RESULT: max_{2<=n<=10000000} M_n = 148, first at n=2886673

The final run used 64.370 seconds wall time on this VM. All arithmetic is

exact integer arithmetic.

5. What remains, precisely

The unresolved lemma is now explicit: prove, for every \(B\), the existence of

arbitrarily large centers \(n\) for which all paired-gap partial sums in

the growing window (8) exceed \(B\).

Known large-gap theorems concern one \(g_n\); bounded-gap or prime-tuples

results control fixed finite patterns. Neither controls the same center

through the \(n^{1-o(1)}\) unconditional window in (8), or even the

\(n^{3/4+o(1)}\) RH window in (9). This is the exact uniformity gap in the

standard machinery, rather than merely “primes are irregular.” **(a) for the

logical requirement; (c) as a diagnosis of available methods**

Extending the finite audit to McNew's \(1.6\times10^8\) index range would

require the first roughly \(3.2\times10^8\) primes, ending near

\(7\times10^9\). A direct 64-bit prime array alone is about 2.56 GB.

An optimized segmented C++ implementation would plausibly cost on the order

of \(0.1\)–\(1\) core-hour plus several GB of memory, depending on compression

and certificate handling. I did not run that here, both because of the

few-CPU-minute limit and because McNew has already computed a larger

distributional range. Such a run could improve the exact record table, but

no finite extension can prove the required limsup.

PARTIAL: Reduced the conjecture to a precise growing paired-gap window and certified every prefix record through \(n=10,000,000\), with maximum \(148\) first at \(n=2,886,673\); unboundedness remains open at the stated uniform-window lemma.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger