This paper studies 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. The authors describe a Lucasian primality test for these numbers i...This paper studies 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. The authors describe a Lucasian primality test for these numbers in certain cases, which runs in deterministic quasi-quadratic time. In particular, the authors construct a Lucasian primality test for numbers of the form 3 · 5 · 17 · 2~n± 1, where n is a positive integer, in half of the cases among the congruences of n modulo 12, by means of a Lucasian sequence with a suitable seed not depending on n. The methods of Bosma(1993), Berrizbeitia and Berry(2004), Deng and Huang(2016) can not test the primality of these numbers.展开更多
Using the idea of Sinnott,Gillard and Schneps,we prove theμ-invariant is zero for the two-variable primitive p-adic L-function constructed by Kang(2012),which arises naturally in the study of Iwasawa theory for an el...Using the idea of Sinnott,Gillard and Schneps,we prove theμ-invariant is zero for the two-variable primitive p-adic L-function constructed by Kang(2012),which arises naturally in the study of Iwasawa theory for an elliptic curve with complex multiplication(CM).展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos.11601202,11401312,11701284the High-Level Talent Scientific Research Foundation of Jinling Institute of Technology under Grant Nos.jit-b-201526,RCYJ201408+1 种基金the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant No.17KJB110004National Key R.and D.Program “Cyberspace Security” Key Special Project under Grant No.2017YFB0802800
文摘This paper studies 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. The authors describe a Lucasian primality test for these numbers in certain cases, which runs in deterministic quasi-quadratic time. In particular, the authors construct a Lucasian primality test for numbers of the form 3 · 5 · 17 · 2~n± 1, where n is a positive integer, in half of the cases among the congruences of n modulo 12, by means of a Lucasian sequence with a suitable seed not depending on n. The methods of Bosma(1993), Berrizbeitia and Berry(2004), Deng and Huang(2016) can not test the primality of these numbers.
基金supported by National Natural Science Foundation of China (Grant No. 11171141)the State Key Development Program for Basic Research of China (973 Program) (Grant No. 2013CB834202)+2 种基金Natural Science Foundation of Jiangsu Province of China (NSFJ) (Grant No. BK2010007)a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD)the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China (Grant No. 708044)
文摘Using the idea of Sinnott,Gillard and Schneps,we prove theμ-invariant is zero for the two-variable primitive p-adic L-function constructed by Kang(2012),which arises naturally in the study of Iwasawa theory for an elliptic curve with complex multiplication(CM).
