Lower Bounds for Least Quadratic Non-Residues

Graham, S. W.; Ringrose, C. J.

doi:10.1007/978-1-4612-3464-7_18

Cited by 94 publications

(119 citation statements)

References 12 publications

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“…For p = 19, by quadratic reciprocity (or directly), the degree of regularity of the graph G is r = For p = 17 the graph G is 2-regular, and is in fact a cycle: (1, 8, 13, 2, 16, 9, 4, 15) (note that all sums of adjacent elements in this cycle are quadratic residues). For p = 23 the set of quadratic residues is Q = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}, the corresponding graph is 4-regular and it is not difficult to find in it a Hamilton cycle: (1, 2,16,8,4,9,3,6,18,13,12).…”

Section: Lemma 22 ([20])mentioning

confidence: 99%

“…More generally, it results from the work of Graham and Ringrose [13] and Iwaniec [18] that (8) holds provided d has prime factors bounded by d δ( ) (see Chang [7] for a unified treatment), except for a possible Siegel zero. Recall that this is a real zero β 0 of an L-function L(s, ψ) (mod d), for a real character ψ, which is close to 1, say…”

Section: Sum-free Subgroupsmentioning

confidence: 99%

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Additive Patterns in Multiplicative Subgroups

Alon

Bourgain

2014

Geom. Funct. Anal.

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The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q 3/4 in the finite field F q must contain an additive relation x + y = z.Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p 1/3 ) in prime fields F p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q 3/4+o(1) of F q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.

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Section: Lemma 22 ([20])mentioning

confidence: 99%

Section: Sum-free Subgroupsmentioning

confidence: 99%

Additive Patterns in Multiplicative Subgroups

Alon

Bourgain

2014

Geom. Funct. Anal.

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“…From the available numerical evidence, the least quadratic nonresidue modulo p appears to grow no more quickly than a small constant times log p [26], The scant evidence on prime testing [35] suggests a bound of perhaps 0(log n) for the least witness to the compositeness of n . Graham and Ringrose [ 18] have shown that the least quadratic nonresidue modulo a prime p is infinitely often Q(logp log log log/? ), and Montgomery [31] gave a bound of f2(log/?loglog/?)…”

Section: Introductionmentioning

confidence: 99%

Explicit bounds for primality testing and related problems

Bach¹

1990

Math. Comp.

170

228

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Abstract.Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is 0(log m). This has been generalized by Lagañas. Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log m .

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“…Indeed, it has been shown in [2] that there exists a constant c > 0 such that for infinitely many primes q the smallest quadratic non-residue modulo q is at least c log q log log log q (under the ERH the same result is known with c log q log log q). Therefore for such q, T = c log q log log log q and any m ≥ 1 we have P m,T (k) = 0 whenever k is one of the (q − 1)/2 quadratic non-residues modulo q.…”

Section: Remarksmentioning

confidence: 95%