On consecutive kth power residues, II

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Adolf Hildebrand, On consecutive $k$th power residues, II, Michigan Mathematical Journal 38 (1991), no. 2, 241–253, DOI 10.1307/mmj/1029004331.

The edition read is a scan of the journal article from the journal's Project Euclid page for the DOI, fourteen pages (the journal pages 241–253 and a blank last page) with no text layer, obtained on 2026-10-16.

For positive integers $k$ and $l$ and a large prime $p$, let $r(k,l,p)$ be the least positive $r$ such that $r, r+1, \dots, r+l-1$ are all $k$th power residues modulo $p$, and let $\Lambda(k,l) = \limsup_{p \to \infty} r(k,l,p)$. The introduction recalls that $\Lambda(k,l) = \infty$ for $l \ge 4$ and every $k \ge 2$, and for $l = 3$ and every even $k$, and that an earlier paper of the author proved $\Lambda(k,2) < \infty$ for prime $k$. Theorem 1 proves $\Lambda(k,2) < \infty$ for every positive integer $k$. It is deduced from Theorem 2: for each $k$ there is a constant $c_0(k)$ such that every completely multiplicative function $f$ whose values are $k$th roots of unity takes the value $f(n) = f(n+1) = 1$ at some $n \le c_0(k)$.

Bears on. Theorem 1 answers the first question of Problem 436, whether $\Lambda(k,2)$ is finite for all $k$; the question for $\Lambda(k,3)$ with odd $k$ is not addressed. Read status: claims checked for Theorems 1 and 2 as stated on p. 241, read clause by clause on the page image; the proofs are unread, and no proof is verified on this card.

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