---

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 (§ 2, printed p. 1716; unnumbered).

• For $2\le x\le1120$, $\varrho^*(x)<\pi(x)$. The paper concludes: "that is, Conjecture B holds for $x\le1120$."
• For $1121\le x\le1426$, $\varrho^*(x)\le\pi(x)$.
• $\varrho^*(1422)=\pi(1422)=223$, with an admissible sequence attaining it given by its residue-class description in Table 3 (p. 1716).

The paper recalls (p. 1714) that Schinzel had shown $\varrho^*(x)\le\pi(x)$ for $x\le146$ from Smith's tables, and that Selfridge had shown Conjecture B for $x\le500$ (reported in Riesel's book). A note on p. 1713 records that, after submission, the authors learned that Dan Gordon and Gene Rodemich had extended the calculation of $\varrho^*(n)$ to $n=1600$.

The result is a finite computation. The paper does not spell out how a bound on $\varrho^*(x)$, which governs the limit superior over shifts, yields Conjecture B at every $y$; its conclusion is quoted above as printed.

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; § 2, pp. 1714--1717, the result on p. 1716. The edition read is identified on the source card.

Read depth. Claims checked: the statements, Tables 1--3 and the algorithm's steps were read on the page images of pp. 1714--1717. The computation was not rerun and nothing here is independently reviewed.

Proof pointer

§ 2, pp. 1714--1716. Restrict to the odd integers of the interval, assume the first one is kept, and for each prime $3\le p_i<x/4$ choose a residue class to erase, indexed by $1\le a_i\le p_i-1$; the resulting sequences are ordered lexicographically by $(a_2,\ldots,a_r)$ (p. 1714). A branch-and-bound search over these choices, the "Algorithm for computing $\varrho^*(x)$", Steps 0--8 (pp. 1714--1716), keeps a lower bound $L$ and abandons a branch once fewer than $L$ elements survive. It gives the exact values of Table 1 (p. 1715) for $x$ up to $1050$, each listed $x$ being, in the paper's words, the length of "the largest interval with an admissible sequence of $\varrho^*(x)$ elements". Table 2 (p. 1716) extends the range to $x\le1120$ by subadditivity, $\varrho^*(x+y)\le\varrho^*(x)+\varrho^*(y)$, the method of Schinzel, for example $\varrho^*(1120)\le\varrho^*(1050)+\varrho^*(70)=187=\pi(1117)$. For $1121\le x\le1426$ the search is rerun with initial lower bound $L=\pi(x)$, so that only sequences of at least $\pi(x)$ points are examined; the run for $x=1422$ took about eleven days (p. 1717).

Dependencies

Subadditivity of $\varrho^*$ (used as in Schinzel, the paper's [5]), and the correctness of the authors' C implementation, which the paper says was made public by ftp.

Bears on

• Problem 855: the result bounds $\varrho^*(x)$ by $\pi(x)$ for every fixed $x$ with $2\le x\le1426$, the range in which no admissible set beats the initial interval. It is finite evidence about small $x$ and says nothing about the problem's regime of large $x$ and $y$.