A note on the second smallest prime 𝑘th power nonresidue

Hudson, Richard H.

doi:10.1090/S0002-9939-1974-0364139-6

A note on the second smallest prime $k$th power nonresidue
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Proc. Amer. Math. Soc. 46 (1974), 343-346 Request permission

Abstract:

Upper bounds for the second smallest prime $k$th power nonresidue, which we denote by ${g_2}(p,k)$, have been given by many authors. Theorem 1 represents an improvement of these bounds, at least for odd $k$. We also give specific estimates for ${g_2}(p,k)$, and an upper bound for the $n$th $(n \geq 2)$ smallest prime $k$th power nonresidue as a function of the first $n - 1$ prime nonresidues. Upper bounds for ${g_2}(p,k)$ should take on new interest since the author has shown elsewhere that the first two consecutive $k$th power nonresidues are bounded above by the product of the first two prime nonresidues.

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