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Primes of the form x 2 + d y 2 with x 0(mod N) or y 0(mod N)

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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

  1. Berrizbeitia P and Iskra B, Gaussian–Mersenne and Eisenstein–Mersenne primes , Math. Comput. 79 (271) (2010) 1779–1791

    Article  MathSciNet  MATH  Google Scholar 

  2. Cohen H and Stevenhagen P, Computational class field theory, in: Algorithmic number theory (eds) J Buhler and P Stevenhagen (2008) (MSRI Publications) vol. 44

  3. Cohen H, A course in computational algebraic number theory (1996) (Berlin: Springer-Verlag)

    Google Scholar 

  4. Cohen H, Advanced topics in computational number theory (2000) (New York: Springer-Verlag)

    Book  MATH  Google Scholar 

  5. Cohen H and Stevenhagen P, Arithmetic of number rings, in: Algorithmic number theory (eds.) J Buhler and P Stevenhagen (2008) (MSRI Publications) vol. 44

  6. Cox D A, Primes of the form x 2 + n y 2 Fermat, class field theory and complex multiplication (1989) (John Wiley & Sons)

  7. Jansen B, Mersenne primes and class field theory, Ph.D. thesis (2012) (Netherlands: Universiteit Leiden)

    Google Scholar 

  8. Jansen B, Mersenne primes of the form x 2 + d y 2, Master’s thesis (2002) (Netherlands: Universiteit Leiden)

  9. Lang S, Algebraic number theory (1994) (New York: Springer-Verlag)

    Book  MATH  Google Scholar 

  10. Lemmermeyer F, Construction of Hilbert 2-class fields, citeseer

  11. Lenstra H W and Stevenhagen P, Mersenne primes and Artin reciprocity, Nieuw Archief voor Wiskunde 5 (1) (2000) 44–54

    MATH  Google Scholar 

  12. Palimar S and Shankar B R, Mersenne primes in real quadratic fields, J. Integer Sequences 15 (5) (2012) 1–12

    MathSciNet  MATH  Google Scholar 

  13. Vaugham T P, The construction of unramified cyclic quartic extension of \(\mathbb {Q}(\sqrt {m})\) , Math. Comput. 45 (171) (1985) 233–242

    MathSciNet  Google Scholar 

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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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Correspondence to SUSHMA PALIMAR.

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Communicating Editor: B Sury

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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

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