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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Power residues and nonresidues in arithmetic progressions


Author: Richard H. Hudson
Journal: Trans. Amer. Math. Soc. 194 (1974), 277-289
MSC: Primary 10A15
MathSciNet review: 0374002
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Abstract: Let k be an integer $ \geq 2$ and p a prime such that $ {v_k}(p) = (k,p - 1) > 1$. Let $ bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c < b,(b,p) = (c,p) = 1)$ be an arithmetic progression. We denote the smallest kth power nonresidue in the progression $ bn + c$ by $ g(p,k,b,c)$, the smallest quadratic residue in the progression $ bn + c$ by $ {r_2}(p,b,c)$, and the nth smallest prime kth power nonresidue by $ {g_n}(p,k),n = 0,1,2, \ldots $.

If $ C(p)$ is the multiplicative group consisting of the residue classes $ \bmod\;p$, then the kth powers $ \bmod\;p$ form a multiplicative subgroup, $ {C_k}(p)$. Among the $ {v_k}(p)$ cosets of $ {C_k}(p)$ denote by T the coset to which c belongs (where c is the first term in the progression $ bn + c)$, and let $ h(p,k,b,c)$ denote the smallest number in the progression $ bn + c$ which does not belong to T so that $ h(p,k,b,c)$ is a natural generalization of $ g(p,k,b,c)$.

We prove by purely elementary methods that $ h(p,k,b,c)$ is bounded above by $ {2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}$ if p is a prime for which either b or $ p - 1$ is a kth power nonresidue. The restriction on b and $ p - 1$ may be lifted if $ p > {({g_1}(p,k))^{7.5}}$. We further obtain a similar bound for $ {r_2}(p,b,c)$ for every prime p, without exception, and we apply our results to obtain a bound of the order of $ {p^{2/5}}$ for the nth smallest prime kth power nonresidue of primes which are large relative to $ \Pi _{j = 1}^{n - 1}{g_j}(p,k)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0374002-7
PII: S 0002-9947(1974)0374002-7
Keywords: kth power residue, arithmetic progressions, subgroup of kth powers, elementary methods, smallest kth power nonresidue in arithmetic progressions
Article copyright: © Copyright 1974 American Mathematical Society