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Claim. Theorem 1.1 of Klarner, J. Algebra 74 (1982), 140--148: if $f_i(x)=a_ix+b_i$ for $i=1,\ldots,r$ are affine maps with positive integer $a_i$ and integer $b_i$ and $\sum_i 1/a_i>1$, then the semigroup they generate is not free on $f_1,\ldots,f_r$, so two distinct words in the $f_i$ define the same map. The problem follows: the words produced by the proof have the same length $k-1$ (a thread comment of 3 December 2025 notes this, and also that two words of different lengths $\ell_1,\ell_2$ can be replaced by the two compositions of length $\ell_1+\ell_2$ taken in either order), and two distinct words of length $k-1$ that agree as maps take the same value at $1$, which is a repeated entry of $A_k$. The problem's hypothesis $a_i,b_i\in\mathbb N$ is a special case of the theorem's.

Postings. The paper, published in January 1982, is not held here; its statement is taken from the account of Kolpakov and Talambutsa, whose Theorem 3 extends it to rational $b_i$ and who report that the proof's second part (the case of equal multipliers) is omitted in the paper and attributed to a reference that leads to an unpublished manuscript. The paper was pointed out in the site's thread on 3 December 2025, and the site's commentary was updated the same day to attribute the first proof to it.

Acceptance. Refereed publication in the Journal of Algebra, and the acceptance of the site's curator, Thomas Bloom, in the commentary, which names this paper as the first proof; the extension by Kolpakov and Talambutsa supplies in print the part the 1982 paper omits, and the independent elementary proof of Barreto reaches the same conclusion. Nothing was reviewed here.

Depends on. No page of this wiki.