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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 (5) (p. 172, proved in Section 3, pp. 175--176). For $k=5$ and $m=2$:

$$\Lambda(5,2)=7888,\qquad p^*=2,\,11,\,41,\,71,\,101 .$$

That is, the exceptional primes for pairs of consecutive quintic residues are exactly $2,11,41,71,101$ (the paper takes this list as known from cyclotomy, p. 175); every other prime $p$ has consecutive quintic residues $r,r+1$ with $r\le7888$; and the bound is attained: the paper shows (p. 176) that there are infinitely many primes $p$ whose least pair of consecutive quintic residues is $(7888,7889)$, where $7888=2^4\cdot17\cdot29$ and $7889=7^3\cdot23$.

Source. D. H. Lehmer, Emma Lehmer and W. H. Mills, Pairs of consecutive power residues, Canadian J. Math. 15 (1963), 172--177: display (5), p. 172; the proof, Section 3, pp. 175--176. The edition read is identified on the source card.

Read depth. Claims checked: the definitions, display (5) and the account of its proof were read clause by clause on the printed pages. The upper bound rests on a computer run whose individual steps the paper does not print, so it was not checked; the attainment step rests on a congruence system (12) whose verification the paper reports doing by machine and by factor tables, and on a theorem cited from another paper, neither checked here.

Proof pointer

Section 3 (pp. 175--176), using the framework of Sections 1 and 2 (pp. 173--175). For a prime $p=kx+1$ and a primitive root, $R(n)$ is the index of $n$ reduced mod $k$, so $n$ is a $k$th power residue exactly when $R(n)=0$. Fixing a finite set $S$ of primes, a prime is classified by the vector of values $R(q)$ for $q\in S$, and a pair $(n,n+1)$ of $S$-smooth numbers not exceeding a bound $L$ disposes of every class for which both members are residues. A machine search over such vectors, organized as a tree of partial "case vectors", is run with $t=22$ and $S$ the first 21 primes together with $101$. Runs with $L=2^{15}$ and $2^{13}$ leave nothing; the run with $L=2^{12}$ leaves one family of vectors, which the pair $(7888,7889)$ disposes of; a final run with $L=7889$, which handled 4568 cases, leaves nothing, giving $\Lambda(5,2)\le7888$. For the lower bound the paper writes down conditions (12) on the values $R(q)$ for the primes $q<7888$ under which the least $n$ with $R(n)=R(n+1)=0$ is $7888$, and obtains infinitely many such primes from Kummer's theorem, cited from the paper it calls the preceding paper.

Dependencies

Kummer's theorem on the existence of infinitely many primes with prescribed $k$th power characters, cited from another paper (p. 176), and the classical determination of the exceptional primes by cyclotomy (p. 175).

Bears on

• Problem 436: the case $k=5$ 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 $7888$ 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.