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Statement

Notation (p. 11): $n_k(p)$ is the least $k$-th power nonresidue of the prime $p$. The paper gives no further convention; in particular it does not say what $n_k(p)$ is for a prime $p$ with $\gcd(k,p-1)=1$, which has no $k$-th power nonresidue.

Conjecture (4) (p. 11, as printed; introduced as likely, not proved):

$$\sum_{p<x}n_k(p)=\bigl(1+o(1)\bigr)\,c_k\,\frac{x}{\log x}.$$

The paper states no range of $k$ and nothing about $c_k$ beyond its dependence on $k$. The English summary (p. 17) words it "It is very likely true that $\sum_{p<x}n_k(p)=(1+o(1))\frac{c_kx}{\log x}$." For $k=2$ the paper proves it, with $c_2=\sum_{j\ge1}p_j/2^j$: equation (3).

The obstruction (p. 11). For $k>2$ the author has no good upper bound for the number of primes $p<x$ with $n_k(p)>A(x)$, where $A(x)\to\infty$ with $x$. The paper says that (4) would follow if that number were shown to be less than

$$c_1\,\frac{x}{\log x\,A(x)^{2+c_2}},$$

and that this had not been done. Here $c_1$ and $c_2$ are unspecified constants; this $c_2$ is not the $c_k$ of (4) at $k=2$.

Source. P. Erdős, Számelméleti megjegyzések, I. (Remarks on number theory, I.; in Hungarian, with Russian and English summaries on p. 17), Mat. Lapok 12 (1961), 10--17; MR 26 #2410, Zbl 0154.294. The definition, conjecture (4) and the obstruction on printed p. 11, the English summary on p. 17; read on the page images of the edition identified on the source card.

Read depth. Claims checked: the definition, the display (4), the sufficient condition and the English summary were read clause by clause on the page images. The sufficiency claim is stated in the paper without proof and was not checked here.

Proof pointer

None: (4) is a conjecture in this paper, proved here only for $k=2$, as equation (3).

Bears on

• Problem 980: (4) is the problem's question, posed here for the least $k$-th power nonresidue with no convention for the primes that have none; the problem page cites it at p. 11 and records how it reads the sum. This paper proves only the case $k=2$.