A000236

[N. J. A. Sloane]

here is the Math Rev of J.'s 1964 paper:"

 
MR0161824 (28 #5028) 
Jordan, J. H. 
Pairs of consecutive power residues or non-residues. 
Canad. J. Math. 16 1964 310--314.
10.06





                                   


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Let $k$ be a positive integer, $p=k+1$ a prime and $g$ a primitive root of $p$. Let $g^{\text{Ind}\,u}\equiv u (\text{mod}\,p)$. 

Recently various authors [the reviewer and E. Lehmer, Proc. Amer. Math. Soc. 13 (1962), 102--106; MR0138582 (25 \#2025); the reviewer, E. Lehmer and W. Mills, Canad. J. Math. 15
(1963), 172--177; MR0146134 (26 \#3660)] have been concerned with the function $\Lambda(k,m)=\max\{r(k,m,p)\}$, where $r(k,m,p)$ denotes the least $r$ for which $$ \multline
\text{Ind}\,(r)\equiv\text{Ind}\,(r+1)\equiv\cdots\equiv\\ \text{Ind}\,(r+m-1)\equiv 0\quad(\text{mod}\,p-1) \endmultline\tag1 $$ and the maximum is taken over all primes $p$ for which $r$
exists. $\Lambda(k,2)$ increases rapidly with $k\colon\Lambda(2,2)=9$, $\Lambda(3,2)=77$, $\Lambda(4,2)=1224$, $\Lambda(5,2)=7888$, $\Lambda(6,2)=202124$,
$\Lambda(7,2)=1649375$. 

The author defines another function $\Lambda^*(k,m)$ obtained by deleting the condition of divisibility in (1) so that one asks only for the first appearance of $m$ consecutive integers whose
$k$th power characters are identical. $\Lambda^*(k,2)$ turns out to be very much smaller. The author has established that $\Lambda^*(2,2)=3$, $\Lambda^*(3,2)=8$, $\Lambda^*(4,2)=20$,
$\Lambda^*(5,2)=44$, $\Lambda^*(6,2)=80$, $\Lambda^*(7,2)=343$. The proofs are short enough to be done "by hand", although the last two proofs are omitted to save space. The further
results $\Lambda^*(2k,3)=\Lambda^*(k,4)=\infty$ are also obtained in the same manner as for $\Lambda$.