Abstract
In this paper, we characterize primes of the form x 2 + d y 2 with x≡0(mod N) or y≡0(mod N) for positive integers N and d with d being square free.
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References
Berrizbeitia P and Iskra B, Gaussian–Mersenne and Eisenstein–Mersenne primes , Math. Comput. 79 (271) (2010) 1779–1791
Cohen H and Stevenhagen P, Computational class field theory, in: Algorithmic number theory (eds) J Buhler and P Stevenhagen (2008) (MSRI Publications) vol. 44
Cohen H, A course in computational algebraic number theory (1996) (Berlin: Springer-Verlag)
Cohen H, Advanced topics in computational number theory (2000) (New York: Springer-Verlag)
Cohen H and Stevenhagen P, Arithmetic of number rings, in: Algorithmic number theory (eds.) J Buhler and P Stevenhagen (2008) (MSRI Publications) vol. 44
Cox D A, Primes of the form x 2 + n y 2 Fermat, class field theory and complex multiplication (1989) (John Wiley & Sons)
Jansen B, Mersenne primes and class field theory, Ph.D. thesis (2012) (Netherlands: Universiteit Leiden)
Jansen B, Mersenne primes of the form x 2 + d y 2, Master’s thesis (2002) (Netherlands: Universiteit Leiden)
Lang S, Algebraic number theory (1994) (New York: Springer-Verlag)
Lemmermeyer F, Construction of Hilbert 2-class fields, citeseer
Lenstra H W and Stevenhagen P, Mersenne primes and Artin reciprocity, Nieuw Archief voor Wiskunde 5 (1) (2000) 44–54
Palimar S and Shankar B R, Mersenne primes in real quadratic fields, J. Integer Sequences 15 (5) (2012) 1–12
Vaugham T P, The construction of unramified cyclic quartic extension of \(\mathbb {Q}(\sqrt {m})\) , Math. Comput. 45 (171) (1985) 233–242
Acknowledgements
The authors would like to thank the referee for carefully reading the manuscript and for helpful suggestions. Special thanks to Professor B R Shankar, Department of MACS, NITK, Surathkal for useful comments on the draft. The second author (SP) is immensely thankful to Prof. Chandan Singh Dalawat, H.R.I. Allahabad, for the kind help and encouragement given to her to understand [11] (during her visit to H.R.I., Allahabad, in December 2011) and to write [12] with Prof. B. R. Shankar, and this further helped her to write the present paper. She is also thankful for the financial support awarded to her by the NBHM – DAE – Post Doctoral Fellowship, Govt. of India.
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DUKKIPATI, A., PALIMAR, S. Primes of the form x 2 + d y 2 with x ≡ 0(mod N) or y ≡ 0(mod N). Proc Math Sci 127, 35–43 (2017). https://doi.org/10.1007/s12044-016-0294-3
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DOI: https://doi.org/10.1007/s12044-016-0294-3
Keywords
- Artin reciprocity
- cyclic quartic unramified extension
- Gaussian–Mersenne prime
- Eisenstein–Mersenne prime