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Statement

Definitions (printed p. 1713). A sequence of integers $b_1<b_2<\cdots<b_k$ is admissible when, for each prime $p$, some residue class modulo $p$ contains none of the $b_i$. The paper sets $\varrho^*(x)$ to be the largest number $k$ of elements of an admissible sequence $y<b_1<b_2<\cdots<b_k\le y+x$ lying in an interval of length $x$, and $\varrho(x)$ to be the limit superior, as the shift tends to infinity, of the number of primes in an interval of length $x$. The print writes the latter as "$\varrho(x)=\limsup_{x\to\infty}(\pi(x+y)-\pi(x))$" [sic], with the roles of $x$ and $y$ crossed; the reading consistent with the rest of the paper is $\varrho(x)=\limsup_{y\to\infty}\bigl(\pi(x+y)-\pi(y)\bigr)$.

Conjecture A (Prime $k$-tuples Conjecture, p. 1713). "Let $b_1<b_2<\cdots<b_k$ be an admissible sequence. Then there exist infinitely many integers $n$ for which $n+b_1,n+b_2,\ldots,n+b_k$ are prime."

Conjecture B (p. 1713). "$\pi(x+y)-\pi(y)\le\pi(x)$."

The paper attributes both conjectures to Hardy and Littlewood (its reference [2], Acta Math. 44 (1923)). Conjecture B is printed with no quantifier on $x$ and $y$; the paper glosses it as saying that no interval of length $x$ holds more primes than the initial interval $[1,x]$. It states that Conjecture A implies $\varrho^*(x)=\varrho(x)$, and recalls, from Hensley and Richards (its reference [3]), that $\varrho^*(x)>\pi(x)$ for all large enough $x$, so that Conjecture B is incompatible with Conjecture A. That theorem is cited here, not proved; its library home is hensley_1974_primes_intervals.

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; § 1, printed p. 1713. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and Conjectures A and B were read clause by clause on the page image of p. 1713. Nothing here is independently reviewed.

Proof pointer

A conjecture and definitions; nothing is proved. The implication from Conjecture A to $\varrho^*(x)=\varrho(x)$ is asserted on p. 1713 without proof.

Dependencies

None within the paper. The incompatibility with Conjecture A rests on the cited Hensley--Richards theorem.

Bears on

• Problem 855: Conjecture B is the problem's inequality $\pi(x+y)\le\pi(x)+\pi(y)$ written for an interval of length $x$ starting at $y$. The problem asks it for large $x$ and $y$; the paper prints it with no quantifier. The function $\varrho^*$ defined here is the quantity the paper's computations bound.