We prove several finite product-sum identities involving the q-binomial coefficient, one of which is used to prove an amazing identity of Gauss. We then use this identity to evaluate certain ...quadratic Gauss sums and, together with known properties of quadratic Gauss sums, we prove the quadratic reciprocity law for the Jacobi symbol. We end our article with a new proof of Jenkins’ lemma, a lemma analogous to Gauss’ lemma. This article aims to show that Gauss’ amazing identity and the properties of quadratic Gauss sums are sufficient to establish the quadratic reciprocity law for the Jacobi symbol.
A new code concept is used for the L1 civil（L1C） signal of the global positioning system（GPS）.The generation of L1C codes is quite different from the generation of traditional ranging codes.Thus,it ...is necessary to find a method for the correct generation to pave the way for future research.L1C codes are based on only one Legendre sequence which consists of Legendre symbols.To calculate these Legendre symbols,the Euler criterion is always used to evaluate quadratic residues.However,due to the great length of L1C codes,this procedure causes overflow problems.Therefore,the quadratic reciprocity law,some related theorems and properties are introduced to solve the problems.Moreover,if the quadratic reciprocity law,some related theorems and properties are used to calculate different Legendre symbols,the combination modes may vary,which causes a complex generation process.The proposed generation method deals with this complex generation process effectively.In addition,through simulations,it is found that the autocorrelation features of obtained Legendre sequences and L1C codes are in accordance with theoretical results,which proves the correctness of the proposed method.
If E is an elliptic curve, then the Galois group of the extension generated by the n-torsion points acts on these points. We prove a quadratic reciprocity law involving this group action. This law is ...an extension of the usual quadratic reciprocity law.
Fast Algorithms on Primality Testing for Numbers 255 ⋅ 2^n ± 1 Huang, Dandan; Zhang, Zheng; Tang, Zhihao
2019 IEEE SmartWorld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computing, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld/SCALCOM/UIC/ATC/CBDCom/IOP/SCI),
2019-Aug.
Conference Proceeding
In this paper, we study the problem of primality testing for numbers of the form h · 2 n ± 1, where h <; 2 n is odd, and n is a positive integer. We describe explicit Lucasian primality tests for ...these numbers in certain cases, which run in deterministic quasi-quadratic time. In particular, we implement the explicit algorithms related to numbers M n = h · 2 n - 1 for the case h =255. And we find large prime numbers which are more than 512 bits efficiently. The algorithms of Bosma (1993), Berrizbeitia and Berry (2004), Deng and Huang (2016) can not generate these prime numbers, even can not test the primality of these numbers.
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