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On the number of zeros of diagonal forms


Author: Tsuneo Ishikawa
Journal: Math. Comp. 64 (1995), 841-854
MSC: Primary 11D79; Secondary 11T22, 11T23
DOI: https://doi.org/10.1090/S0025-5718-1995-1284669-4
MathSciNet review: 1284669
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Abstract: In this paper, we study the number of zeros of the equation $ X_1^e + X_2^e + \cdots + X_n^e \equiv 0\; \pmod p$, where $ e > 1$ is a positive integer and $ p \equiv 1\; \pmod e$ is a prime. For $ e = 5,7$, we compute the recurrence satisfied by these numbers, using the generalized Jacobi sums.


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  • [1] L.D. Baumert and H. Fredricksen, The cyclotomic numbers of order eighteen with applications to difference sets, Math. Comp. 21 (1967), 204-219. MR 0223322 (36:6370)
  • [2] B.C. Berndt and R.J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J.Number Theory 11 (1979), 349-398. MR 544263 (81j:10054)
  • [3] S. Chowla, J. Cowles and M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory 9 (1977), 502-506. MR 0453632 (56:11894)
  • [4] L.E. Dickson, Cyclotomy higher congruences, and Waring's problem I,II, Amer. J. Math. 57 (1935), 463-474. MR 1507087
  • [5] K. Ireland and M. Rosen, A classical introduction to modern number theory, Graduate Texts in Math., vol. 84, Springer-Verlag, New York, Heidelberg, Berlin, 1982. MR 661047 (83g:12001)
  • [6] P.A. Leonard and K.S. Williams, The cyclotomic numbers of order seven, Proc. Amer. Math. Soc. 51 (1975), 295-300. MR 0371868 (51:8085)
  • [7] -, The cyclotomic numbers of order eleven, Acta Arith. 26 (1975), 366-383. MR 0376513 (51:12688)
  • [8] J.B. Muskat, The cyclotomic numbers of order fourteen, Acta Arith. 11 (1966), 263-279. MR 0193081 (33:1302)
  • [9] -, On Jacobi sums of certain composite order, Trans. Amer. Math. Soc. 134 (1969), 483-502. MR 0232752 (38:1075)
  • [10] J.B. Muskat and A.L. Whiteman, The cyclotomic numbers of order twenty, Acta Arith. 17 (1970), 185-216. MR 0268151 (42:3050)
  • [11] G. Myerson, On the number of zeros of diagonal cubic forms, J. Number Theory 11 (1979), 95-99. MR 527764 (80e:12023)
  • [12] D.R. Richman, Some remarks on the number of solutions to the equation $ f({X_1}) + \cdots + f({X_n}) = 0$, Stud. Appl. Math. 71 (1984), 263-266. MR 769080 (86d:11100)
  • [13] A.L. Whiteman, The cyclotomic numbers of order sixteen, Trans. Amer. Math. Soc. 86 (1957), 401-413. MR 0092807 (19:1160h)
  • [14] -, The cyclotomic numbers of order ten, Proc. Sympos. Appl. Math., vol. 10, Amer. Math. Soc., Providence, RI, 1960, pp. 95-111. MR 0113851 (22:4682)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1284669-4
Article copyright: © Copyright 1995 American Mathematical Society

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