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Claim. The first question of Problem 436 has answer yes: $\Lambda(k,2)<\infty$ for every $k$. The result is Theorem 1 of A. Hildebrand, On consecutive $k$th power residues. II, Michigan Math. J. 38 (1991), no. 2, 241--253, DOI 10.1307/mmj/1029004331 (received 28 March 1990). The paper defines $r(k,l,p)$ as the least $r$ such that $r,r+1,\ldots,r+l-1$ are all $k$th power residues modulo $p$, which exists for every large prime $p$ by a theorem of Brauer, and $\Lambda(k,l)=\limsup_{p\to\infty}r(k,l,p)$, the problem's definition. Theorem 1 (p. 241) states that $\Lambda(k,2)<\infty$ for all positive integers $k$, and the paper restates it: for each $k$ there is a constant $c_0(k)$ such that every sufficiently large prime $p$ has a pair $r,r+1$ of consecutive $k$th power residues with $1\le r\le c_0(k)$. No explicit bound for $c_0(k)$ is given.

The proof's shape. Theorem 1 is deduced (p. 242) from the paper's Theorem 2: for each $k$ there is a constant $C_0(k)$ such that every completely multiplicative function $f$ on the positive integers with $f^k=1$ has a positive integer $n\le C_0(k)$ with $f(n)=f(n+1)=1$. The deduction reduces to primes $p\equiv1\pmod k$, since the $k$th power residues modulo $p$ are the $d$th power residues for $d=(k,p-1)$, and builds $f$ from a primitive root modulo $p$ so that $f(n)=1$ exactly when $n$ is a $k$th power residue. The proof of Theorem 2 finds a set of integers in which the quotients of any two members by their greatest common divisor are consecutive, Heath-Brown's special sets, on which $f$ takes the value $1$, using the pigeonhole principle, Ramsey's theorem, elementary sieve estimates and estimates for multiplicative functions, as the introduction lists. Part I of the paper (Monatsh. Math. 102 (1986), 103--114) proved the case of prime $k$; this page covers that case, so Part I has no page of its own. Before the theorem, $\Lambda(k,2)$ was known finite for $k\le7$ by machine computation, with the exact values the problem page lists.

Covers. The first question, answered yes: $\Lambda(k,2)$ is finite for every $k\ge2$. It does not cover the second question, whether $\Lambda(k,3)$ is finite for every odd $k$, or the third, how $\Lambda(k,2)$ and $\Lambda(k,3)$ grow with $k$.

Depends on. No page of this wiki.

Acceptance. Refereed: the Michigan Mathematical Journal, a refereed journal, cited with its venue above. The site's commentary credits the theorem with the first question, but 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 proof of Theorem 2 (sections 2 to 4 of the paper) carries no independent review in this corpus.