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Statement

Setting (p. 172). For positive integers $k$ and $m$ and a prime $p$, $r(k,m,p)$ is the least $r$ such that the $m$ consecutive positive integers $r,r+1,\ldots,r+m-1$ are all $k$th power residues of $p$. For fixed $k$ and $m$ a prime $p^*$ is exceptional when no $m$ consecutive integers are all $k$th power residues of $p^*$, and $\Lambda(k,m)$ is the maximum of $r(k,m,p)$ over all non-exceptional primes $p$.

Result (6) (p. 172, proved in Section 4, pp. 176--177). For $k=6$ and $m=2$:

$$\Lambda(6,2)=202124,\qquad p^*=2,\,3,\,5,\,7,\,13,\,19,\,43,\,61,\,97,\,157,\,277 .$$

The paper glosses it (p. 172, quoted): "every prime $p>277$ has two consecutive numbers which are sextic residues and do not exceed 202125." It adds that the limit is best possible, since infinitely many primes have $(202124,202125)$ as their least pair of consecutive sextic residues; this is established on p. 177.

Source. D. H. Lehmer, Emma Lehmer and W. H. Mills, Pairs of consecutive power residues, Canadian J. Math. 15 (1963), 172--177: display (6), p. 172; the proof, Section 4, pp. 176--177. The edition read is identified on the source card.

Read depth. Claims checked: the definitions, display (6), its gloss and the account of its proof were read clause by clause on the printed pages. The paper describes the proof only as a computer run by "a similar, but more complicated procedure" than for $k=5$ and does not print its steps, so it was not checked; the attainment step rests on the congruence system (13) and a theorem cited from another paper, neither checked here.

Proof pointer

Section 4 (pp. 176--177), by the method of Sections 1--3 described on the page for display (5): a machine search over the vectors of sextic characters $R(q)$ of a fixed finite set of small primes, disposing of each class by a pair of consecutive smooth numbers that are both sextic residues for it. The final run, with $L=202125$, considered 25411 case vectors. For the lower bound the paper gives conditions (13) on $R(q)$ for the primes $q\le202123$ that make $(202124,202125)$ the least pair, and obtains infinitely many such primes from Theorem 3 of the paper it calls the preceding paper.

Dependencies

Theorem 3 of another paper, on primes with prescribed vectors of power characters (p. 177), and the framework shared with display (5).

Bears on

• Problem 436: the case $k=6$ of the first question, whether $\Lambda(k,2)$ is finite, with its exact value. The problem defines $\Lambda(k,m)$ as $\limsup_{p\to\infty} r(k,m,p)$ while the paper takes the maximum over non-exceptional primes; since infinitely many primes attain $202124$ and none exceeds it, the two agree here (an observation of this page, not of the paper). A single value says nothing about the growth of $\Lambda(k,2)$ in $k$, which the third question asks about.