Power residues and nonresidues in arithmetic progressions

Author:
Richard H. Hudson

Journal:
Trans. Amer. Math. Soc. **194** (1974), 277-289

MSC:
Primary 10A15

DOI:
https://doi.org/10.1090/S0002-9947-1974-0374002-7

MathSciNet review:
0374002

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Abstract: Let *k* be an integer and *p* a prime such that . Let be an arithmetic progression. We denote the smallest *k*th power nonresidue in the progression by , the smallest quadratic residue in the progression by , and the *n*th smallest prime *k*th power nonresidue by .

If is the multiplicative group consisting of the residue classes , then the *k*th powers form a multiplicative subgroup, . Among the cosets of denote by *T* the coset to which *c* belongs (where *c* is the first term in the progression , and let denote the smallest number in the progression which does not belong to *T* so that is a natural generalization of .

We prove by purely elementary methods that is bounded above by if *p* is a prime for which either *b* or is a *k*th power nonresidue. The restriction on *b* and may be lifted if . We further obtain a similar bound for for every prime *p*, without exception, and we apply our results to obtain a bound of the order of for the *n*th smallest prime *k*th power nonresidue of primes which are large relative to .

**[1]**Alfred Brauer,*Über den kleinsten quadratischen Nichtrest*, Math. Z.**33**(1931), no. 1, 161–176 (German). MR**1545207**, https://doi.org/10.1007/BF01174349**[2]**Alfred Brauer,*Über die Verteilung der Potenzreste*, Math. Z.**35**(1932), no. 1, 39–50 (German). MR**1545287**, https://doi.org/10.1007/BF01186547**[3]**Alfred Brauer,*On the non-existence of the Euclidean algorithm in certain quadratic number fields*, Amer. J. Math.**62**(1940), 697–716. MR**0002994**, https://doi.org/10.2307/2371480**[4]**Ezra Brown,*A theorem on biquadratic reciprocity*, Proc. Amer. Math. Soc.**30**(1971), 220–222. MR**0280462**, https://doi.org/10.1090/S0002-9939-1971-0280462-5**[5]**D. A. Burgess,*A note on the distribution of residues and non-residues*, J. London Math. Soc.**38**(1963), 253–256. MR**0148628**, https://doi.org/10.1112/jlms/s1-38.1.253**[6]**P. Erdös and Chao Ko,*Note on the Euclidean algorithm*, J. London Math. Soc.**13**(1938), 3-8.**[7]**Loo-Keng Hua,*On the distribution of quadratic nonresidues and the Euclidean algorithm in real quadratic fields. I*, Trans. Amer. Math. Soc.**56**(1944), 537–546. MR**0011481**, https://doi.org/10.1090/S0002-9947-1944-0011481-X**[8]**Richard H. Hudson,*On squences of consecutive quadratic nonresidues*, J. Number Theory**3**(1971), 178–181. MR**0274385**, https://doi.org/10.1016/0022-314X(71)90034-5**[9]**-,*On the distribution of k-th power non-residues*, Duke Math. J.**39**(1972), 85-88.**[10]**Richard H. Hudson,*Prime 𝑘-th power non-residues*, Acta Arith.**23**(1973), 89–106. MR**0321849**, https://doi.org/10.4064/aa-23-1-89-106**[11]**Richard H. Hudson,*On the distribution of 𝑘-th power non residues in the interval [1,𝑝^{𝑎}],2/5<𝑎≤4/9*, J. Reine Angew. Math.**260**(1973), 178–180. MR**0316362**, https://doi.org/10.1515/crll.1973.260.178**[12]**C. Stengel,*Über quadratische Nichtreste von der Form*, J. Reine Angew. Math.**153**(1924), 208-214.**[13]**Clifton T. Whyburn,*The second smallest quadratic non-residue*, Duke Math. J.**32**(1965), 519–528. MR**0180524**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0374002-7

Keywords:
*k*th power residue,
arithmetic progressions,
subgroup of *k*th powers,
elementary methods,
smallest *k*th power nonresidue in arithmetic progressions

Article copyright:
© Copyright 1974
American Mathematical Society