A generalization of Scholz’s reciprocity law

Budden, Mark; Eisenmenger, Jeremiah; Kish, Jonathan

doi:10.5802/jtnb.604

Cited by 3 publications

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“…Proof. One easily observes that the pairs (3, 1) and (1,3) are the only solutions to the congruence N ≡ XY mod 4. Therefore, one of the prime factors of N is congruent to 3 modulo 4, whereas the other one is congruent to 1 modulo 4.…”

Section: The Quadratic Residuosity Problemmentioning

confidence: 99%

“…Burde's and Scholz's reciprocity laws for the biquadratic case and their generalizations for the octic and for higher power cases are well-known. We refer the reader to [5], to [1,Theorem 3] and to [3,Theorem 3.1]. For a general overview on power residue symbols in number fields and other rational power residue symbols, we suggest the survey [6].…”

Section: Introductionmentioning

confidence: 99%

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Computational aspects of rational residuosity

Hittmeir

2018

Int. J. Number Theory

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In this paper, we consider an extension of Jacobi's symbol, the so called rational $2^k$-th power residue symbol. In Section 3, we prove a novel generalization of Zolotarev's lemma. In Sections 4, 5 and 6, we show that several hard computational problems are polynomial-time reducible to computing these residue symbols, such as getting nontrivial information about factors of semiprime numbers. We also derive criteria concerning the Quadratic Residuosity Problem.Comment: 16 page

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Section: The Quadratic Residuosity Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computational aspects of rational residuosity

Hittmeir

2018

Int. J. Number Theory

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“…It is easy to check that β = 11+ Remark 4. Budden, Eisenmenger & Kish [2] have generalized Scholz's reciprocity law to higher powers; can the reciprocity law (λ p /q) = (λ q /p) proved above also be generalized in this direction?…”

Section: Additional Remarksmentioning

confidence: 99%

A supplement to Scholz's reciprocity law

Lemmermeyer¹

2007

Acta Arith.

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Rational residuacity of primes

Budden

Collins

Lea

et al. 2010

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The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of ‫(ޑ‬ζ p ). As an example, a new generalization of Burde's law is provided.

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A generalization of Scholz’s reciprocity law

Cited by 3 publications

References 15 publications

Computational aspects of rational residuosity

Computational aspects of rational residuosity

A supplement to Scholz's reciprocity law

Rational residuacity of primes

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