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Claim. The case $k=3$ of the second question of Problem 436 is settled: $\Lambda(3,3)=23532$, in particular finite. The result is Theorem 1 of D. H. Lehmer, E. Lehmer, W. H. Mills and J. L. Selfridge, Machine proof of a theorem on cubic residues, Math. Comp. 16 (1962), no. 80, 407--415, DOI 10.1090/s0025-5718-1962-0162379-2 (received 3 April 1962). The paper calls three consecutive positive integers a triplet and calls a prime exceptional when it has no triplet of cubic residues; every prime $p>3$ with $p\equiv2\pmod3$ has every number below $p$ as a cubic residue, so apart from $2$ and $3$ only primes $p\equiv1\pmod6$ can be exceptional, and a theorem of Brauer from 1928 had shown that there are only finitely many. Theorem 1 (p. 407) has three parts: (a) the exceptional primes are exactly $2$, $3$, $7$, $13$, $19$, $31$, $37$, $43$, $61$, $67$, $79$, $127$ and $283$; (b) every other prime has a triplet of cubic residues not exceeding $(23532,23533,23534)$; (c) infinitely many primes have that triplet as their smallest. So $r(3,3,p)\le23532$ for every prime outside the finite list, with equality for infinitely many $p$, which is $\Lambda(3,3)=23532$.

The proof's shape. For a prime $p\equiv1\pmod6$ the nonzero residues fall into the cubic residues and two classes of non-residues, and whether an integer whose prime factors lie in a fixed finite set $S$ of primes is a cubic residue depends only on the classes of the members of $S$, a vector modulo $3$ attached to $p$. A triplet whose members factor over $S$ disposes of every vector orthogonal to its three factorization vectors, and a theorem of Kummer gives infinitely many primes for every vector. The machine search found a set $S$ and triplets up to $(23532,23533,23534)$ disposing of every vector not belonging to an exceptional prime, which gives (a) and (b), and a vector whose smallest disposing triplet is that one, which gives (c). The authors report that seven values of the endpoint were tried before $23533$ was found, and their remark records the referee's view that the proof is a machine-aided case analysis rather than a theorem-proving program's output.

Covers. The case $k=3$ of the second question: $\Lambda(3,3)$ is finite, with the value $23532$. It does not cover odd $k\ge5$, nor the growth of $\Lambda(k,2)$ or $\Lambda(k,3)$ in $k$.

Depends on. No page of this wiki.

Acceptance. Refereed: Mathematics of Computation, a refereed journal, cited with its venue above. The site added the credit to its commentary after a comment of 24 October 2025 on its discussion thread (page last edited 25 October 2025); the site labels the problem OPEN, so the credit is not reviewed evidence. The page is dated by the publication year, since the record gives only the year. The machine case analysis carries no independent review in this corpus.