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Statement

Notation: $\varrho^*(x)$ is the largest number of elements of an admissible sequence in an interval of length $x$, as defined on the page for Conjecture B.

Result (§ 3, printed p. 1717; unnumbered). The authors exhibit

• an admissible sequence of $715$ points in an interval of length $5380$, while $\pi(5380)=708$;
• an admissible subsequence of it of $657$ points in an interval of length $4916$, while $\pi(4916)=656$, given by its residue-class description in Table 4 (p. 1717).

Hence $\varrho^*(5380)\ge715>\pi(5380)$ and $\varrho^*(4916)\ge657>\pi(4916)$. The paper sets these against the earlier examples it recalls on p. 1717: Hensley, Richards and Stenberg's $\varrho^*(20000)>\pi(20000)$, and Vehka and Richards's admissible sequence of $1412$ points in length $11763$, where $\pi(11763)=1409$.

The result is unconditional and concerns admissible sequences, not primes. By the prime $k$-tuples conjecture (Conjecture A, p. 1713), each such sequence would have infinitely many translates consisting of primes, giving intervals of length $4916$ with more primes than $[1,4916]$; the paper does not prove that any such translate exists.

Source. David A. Clark and Norman C. Jarvis, "Dense admissible sequences," Mathematics of Computation 70(236) (2001), 1713--1718, https://doi.org/10.1090/s0025-5718-01-01348-5; § 3, p. 1717. The edition read is identified on the source card.

Read depth. Claims checked: the statements and Table 4 were read on the page image of p. 1717. The admissibility of the sequences was not recomputed and nothing here is independently reviewed.

Proof pointer

§ 3, p. 1717. The authors pick lengths $x$ with $\pi(x)-\operatorname{Li}(x)$ small, try every combination of erased residue classes for the first $n$ primes, and for each later prime erase the class removing the fewest surviving elements. With $n=9$ this gave the 715-point sequence in about nine days of computation; a search inside it for a denser shorter stretch gave the 657-point subsequence.

Dependencies

None beyond the computation.

Bears on

• Problem 855: under the prime $k$-tuples conjecture, Table 4 gives $\pi(y+4916)-\pi(y)\ge657>\pi(4916)$ for infinitely many $y$, and the 715-point sequence an excess of at least seven at length $5380$. With $x$ fixed, this contradicts the problem's inequality only if its threshold for "large" is below these lengths; it is a conditional violation at fixed lengths, not an asymptotic counterexample.