Power Nonresidues Research Articles

Euler’s criterion for eleventh power nonresidues

Let p be a prime $$\equiv 1 \!\!\!\pmod {11}$$ . If an integer D with $$(p,D)=1$$ is an eleventh power nonresidue $$\pmod {p}$$ , then $$D^{(p-1)/11} \equiv \alpha \, \!\!\!\pmod {p}$$ for some eleventh root of unity $$\alpha (\not \equiv 1)\,\!\!\!\pmod {p}$$ . In this paper, we establish an explicit expression for $$\alpha $$ in terms of a particular solution of certain quadratic partition of p. Euler’s criterion for eleventh power residues and nonresidues is given with explicit results for $$D=\displaystyle {2,7,11}$$ .

Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing k k th power nonresidues in finite fields F p n \mathbf {F}_{p^n} , where p p is prime and k k is a prime divisor of p n − 1 p^n-1 . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n n and p → ∞ p \rightarrow \infty , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if k k is exponentially large. More generally, assuming the ERH, in time ( n log ⁡ p ) O ( n ) (n \log p)^{O(n)} we can construct a set of elements that generates the multiplicative group F p n ∗ \mathbf {F}_{p^n}^* . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.

On principal ideal testing in algebraic number fields

Let F be an algebraic number field of degree n over ℚ (the rationals). An algorithm is presented for determining whether or not a given ideal in F is principal. This algorithm is applied to the problem of determining the cyclotomic numbers of order 7 for a prime p ≡ 1 (mod 7). Given a 7th power non-residue of p , these numbers can be efficiently computed in O ((log p ) 3 ) binary operations.

A note on prime k-th power nonresidues

Bounds for the second and third smallestprime k-th power nonresidues of odd primes p have been given by Alfred Brauer, Clifton Whyburn, and L. K. Hua. Bounds for the n-th prime residue, n≥4, do not appear in the literature and it would be difficult to obtain bounds as sharp as p1/4 if n is large and k is small. In this note we use the character sum estimates of D. A. Burgess to show that there are on the order of log p/log log pprime k-th power nonresidues less than p1/4 +∈ for every ∈>0 and sufficiently large p.

A sharper bound for the least pair of consecutive k-th power non-residues of non-principal characters (mod p) of order k > 3

A sharper bound for the least pair of consecutive k-th power non-residues of non-principal characters (mod p) of order k > 3

A bound for the first k-1 consecutive k-th power non-residues (mod p)

A bound for the first k-1 consecutive k-th power non-residues (mod p)

A Note on the Second Smallest Prime k th Power Nonresidue

A Note on the Second Smallest Prime k th Power Nonresidue

A note on the second smallest prime 𝑘th power nonresidue

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

Power residues and nonresidues in arithmetic progressions

Let k be an integer ≥ 2 \geq 2 and p a prime such that v k ( p ) = ( k , p − 1 ) > 1 {v_k}(p) = (k,p - 1) > 1 . Let b n + c ( n = 0 , 1 , … ; b ≥ 2 , 1 ≤ c > b , ( b , p ) = ( c , p ) = 1 ) 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 b n + c bn + c by g ( p , k , b , c ) g(p,k,b,c) , the smallest quadratic residue in the progression b n + c bn + c by r 2 ( p , b , c ) {r_2}(p,b,c) , and the nth smallest prime kth power nonresidue by g n ( p , k ) , n = 0 , 1 , 2 , … {g_n}(p,k),n = 0,1,2, \ldots . If C ( p ) C(p) is the multiplicative group consisting of the residue classes mod p \bmod \;p , then the kth powers mod p \bmod \;p form a multiplicative subgroup, C k ( p ) {C_k}(p) . Among the v k ( p ) {v_k}(p) cosets of C k ( p ) {C_k}(p) denote by T the coset to which c belongs (where c is the first term in the progression b n + c ) bn + c) , and let h ( p , k , b , c ) h(p,k,b,c) denote the smallest number in the progression b n + c bn + c which does not belong to T so that h ( p , k , b , c ) h(p,k,b,c) is a natural generalization of g ( p , k , b , c ) g(p,k,b,c) . We prove by purely elementary methods that h ( p , k , b , c ) 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 {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 p - 1 is a kth power nonresidue. The restriction on b and p − 1 p - 1 may be lifted if p > ( g 1 ( p , k ) ) 7.5 p > {({g_1}(p,k))^{7.5}} . We further obtain a similar bound for r 2 ( p , b , c ) {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 {p^{2/5}} for the nth smallest prime kth power nonresidue of primes which are large relative to Π j = 1 n − 1 g j ( p , k ) \Pi _{j = 1}^{n - 1}{g_j}(p,k) .

On the Least k th Power Non-Residue in an Algebraic Number Field

On the Least <i>k</i> th Power Non-Residue in an Algebraic Number Field

The distribution of p­th power non-residues

The distribution of p­th power non-residues

Pairs of Consecutive Power Residues or Non-Residues

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that