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Statement

Setting (pp. 447--448). $P=\{1,2,3,\ldots\}$, $N=\{0,1,2,\ldots\}$, and $\langle mx+ny:1\rangle$ is the least subset of $P$ containing $1$ and closed under $(x,y)\mapsto mx+ny$.

Conjecture 2 (p. 457, quoted). "The set $\langle mx+ny:1\rangle$ contains an infinite arithmetic progression for all $m,n\in P$."

Status in the paper

The paper motivates the conjecture (p. 457) by a statement it says can be proved along the lines of Theorem 4 (no proof is printed): if $a,d,r-1,m_1,\ldots,m_r\in P$ with $(a,d)=(m_1,\ldots,m_r)=1$ and $a+dN\subseteq\langle m_1x_1+\cdots+m_rx_r:1\rangle=S$, then $S$ is a per-set. It adds, in parentheses, that the conjecture has now been proved, citing the authors' reference [2]; that proof is not in this paper.

The paper also records (p. 457) that the conjecture would not make $\langle mx+ny:1\rangle$ a per-set in general: it reports that R. Graham has shown $\langle 3x+3y:1\rangle$ is not a near per-set, though it contains $36+45N$. It then cites "Theorem 12", giving arbitrarily long arithmetic progressions in $\langle mx+ny:1\rangle$ for all $m,n\in P$, as evidence for the conjecture (pp. 457--458); no Theorem 12 is printed in the paper, whose last numbered results are Theorem 11 and Lemma 5 (pp. 460--463).

Proof pointer

None in this paper; the paper attributes the proof to its reference [2].

Read depth

Claims checked: the conjecture and the surrounding remarks on pp. 457--458 were read on the page images of the print, and pp. 458--463 were read there to confirm that no Theorem 12 follows Theorem 11. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. D. A. Klarner and R. Rado, Arithmetic properties of certain recursively defined sets, Pacific J. Math. 53 (1974), no. 2, 445--463, doi:10.2140/pjm.1974.53.445; the edition read is named on the source card.

Bears on

None of the problem pages directly.