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Statement

Setting (p. 216). $\rho^*(x)$ is the size of the largest admissible set in $[1,x]$, a set being admissible when it misses at least one residue class modulo every prime.

Computational results (pp. 223--224, unnumbered). The paper reports:

• By exhaustive search, the first $x$ with $\rho^*(x)=\pi(x)$ is $x=1417$ (p. 223). The paper notes that Jarvis found an admissible set for $x=1422$ with the same cardinality, and that $\pi(x)$ then pulls ahead again.
• The search was continued up to $x=1663$ (p. 223). Figure 3 (p. 224) plots $\rho^*(x)-\pi(x)$ for $x\le1631$, and p. 217 says the paper computes $\rho^*(x)$ for $x<1631$.
• Schinzel's inequality (5), $\rho^*(x+y)\le\rho^*(x)+\rho^*(y)$, combined with the computed values, gives $\rho^*(x)\le\pi(x)$ for $x\le1731$ (p. 223).
• A search found no admissible set of length $1971$ with $298$ elements (pp. 223--224).
• The paper reports Jarvis's result, from a mix of exhaustive search on small primes and a greedy choice for larger primes, that $\rho^*(4930)\ge658>\pi(4930)$ (p. 224).

Before Section 3, the paper records (p. 217) Selfridge's computation that $\rho^*(x)<\pi(x)$ for $x\le500$ and Jarvis's that $\rho^*(x)<\pi(x)$ for $x\le1120$.

Method pointer

Pp. 222--223. The search sieves by primes up to $29$ over one period $[1,P(29)]$, using the Chinese remainder theorem to replace residue choices by translated intervals, and for each promising interval exhausts residue classes modulo $31,37,\ldots$ until too few survivors remain or the next prime exceeds their number. It uses the reflection symmetry, $\rho^*(x)=\rho^*(x-1)$ for even $x$, $\rho^*(x-2)\le\rho^*(x)\le\rho^*(x-2)+1$, and Theorem 1 to prune. Two independently written programs, in C and Fortran, gave the same answers.

Read depth

Claims checked: the reported values were read on the page images of the copy named on the source card. The computations were not repeated here. Nothing here is independently reviewed.

Dependencies

Theorem 1 of the same paper, used for pruning. External inputs named by the paper: Schinzel's inequality (5) and Jarvis's thesis.

Source. Daniel M. Gordon and Gene Rodemich, "Dense admissible sets," Algorithmic Number Theory, Lecture Notes in Computer Science 1423 (1998), 216--225, doi:10.1007/BFb0054864. Pages are the published pagination, pp. 223--224 being pp. 8--9 of the copy read, as the source card explains.

Bears on

• Problem 1204: $\rho^*(x)$ is the largest $k$ with $A(k)\le x-1$, so these values are finite data on $A(k)$: for instance $\rho^*(1417)=\pi(1417)=223$ says $A(223)\le1416$. They do not touch the asymptotic question $A(k)\sim k\log k$ or $B(k)$.