DOI QR코드

DOI QR Code

CONGRUENCE EQUATIONS OF axi + byj ≡ c AND axi + byj + dzt ≡ c(modp) WHEN p=2q+1 WITH p AND q ODD PRIMES

  • KIM, DAE-YEOUL (Department of Mathematics Chonbuk National University) ;
  • KOO, JA-KYUNG (Department of Mathematics Korea Advanced Institute of Science and Technology) ;
  • KIM, MYUNG-HWAN (Department of Mathematical Science Seoul National Univ.)
  • Published : 2005.07.01

Abstract

Let p and q be odd primes with p=2q+1. We study the number of solutions of congruence equations $ax^i\;+\;by^j\;{\equiv}\;c$ (mod p) and a$ax^i\;+\;by^j\;+\;dz^t\;{\equiv}\;c(modp)$

Keywords

References

  1. H. Akazawa, The congruence relations of the number of Fp-rational points on $y^2 = x(x^2 + 2x + 2)$, Math. J. Okayama Univ. 42 (2000), 67-71
  2. B. C. Berndt, Sums of Gauss, Jacobi and Jacobsthal, J. Number Theory 11 (1979), 349-398 https://doi.org/10.1016/0022-314X(79)90008-8
  3. B. C. Berndt and R. C. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math. 23 (1979), 374-437
  4. L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math. 57 (1935), 391-424 https://doi.org/10.2307/2371217
  5. G. Eisenstein, Beitriige zur Kreisteilung, J. Reine Angew. Math. (1844), 269-278
  6. K. Ireland and M. Rosen, A Classical Introduction to Mordern Number Theory, Springer-Verlag, 1981
  7. C. Jacobi, Uber die Kreisteilung ... , J. Reine Angew. Math. (1846), 254-274
  8. I. Niven, H. S. Zuckerman, and H. L. Montgomery, An introduction to the theory of numbers, John Wiley & Sons, Inc., 1991
  9. A. Pekin and H. Icscan, On the solvability of the equation $x^2-py^2={\pm}q$ and the class number of $Q(\sqrt{p})$ for the $p=[(2n+1)q]^2{\pm}1$, Adv. Stud. Contemp. Math. (2004), 87-92, 254-274.
  10. A. R. Rajwade, A note on the number of solutions Np of the congruence $y^2{\equiv}x^3-Dx$ (mod p), Proc. Cambridge Phil. Soc. 67 (1970), 603-605 https://doi.org/10.1017/S0305004100045916
  11. A. R. Rajwade, On rational primes p congruent to 1 (mod 3 or 5), Proc. Cambridge Phil. Soc. 66 (1969), 61-70 https://doi.org/10.1017/S0305004100044716
  12. A. R. Rajwade, On the congruence $y^2{\equiv}x^5$ - a (mod p), Proc. Cambridge Phil. Soc 74 (1973), 473-475 https://doi.org/10.1017/S0305004100077239
  13. J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986
  14. Z. H. Sun, On the theory of cubic residues and non-residues, Acta Arith. 84 (1998), 291-335 https://doi.org/10.4064/aa-84-4-291-335
  15. Z. H. Sun, Supplements to the theory of quartic residues, Acta Arith. 97 (2001), 361-377 https://doi.org/10.4064/aa97-4-5
  16. Surjit Singh and Rajwade, The number of solutions of the congruence $y^2{\equiv}x^4$ - a (mod p), L'Enseignement Math. (1974), 265-263
  17. A. Weil, Number of solutions of equations in a finite field, Bull. Amer. Math. oc. 55 (1949), 497-508
  18. K. S. Williams, A quadratic partition of primes ${\equiv}$ 1 (mod 7), Math. Camp. 28 (1974),1133-1136 https://doi.org/10.2307/2005371
  19. K. S. Williams, Elementary treatment of quadratic partition of primes ${\equiv}$ 1 (mod 7), Illinois J. Math. 18 (1974), 608-621