Products and sums of powers of binomial coefficients mod and solutions of certain quaternary Diophantine systems
Author:
Richard H. Hudson
Journal:
Math. Comp. 43 (1984), 603613
MSC:
Primary 11D09; Secondary 11E20
MathSciNet review:
758208
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Abstract: In this paper we prove that certain products and sums of powers of binomial coefficients modulo , , are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field . Let the four cosets of the subgroup A of quartic residues be given by , and let Let and denote the smallest and next smallest of the respectively. We give new, and unexpectedly simple determinations of and , in terms of the parameters x in the above partition of , in the complicated case that arises when the class number of K is and .
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 [1]
 Duncan A. Buell & Richard H. Hudson, "Solutions of certain quaternary quadratic systems," Pacific J. Math., v. 114, 1984, pp. 2345. MR 755481 (87e:11033)
 [2]
 L. E. Dickson, "Cyclotomy and trinomial congruences," Trans. Amer. Math. Soc., v. 37, 1935, pp. 363380. MR 1501791
 [3]
 C. F. Gauss, "Theoria residuorum biquadraticorum, Comment. I," Comment, soc. reg. sci. Gottingensis rec., v. 6, 1828, p. 27. (Werke vol. 2, p.90.)
 [4]
 Reinaldo E. Giudici, Joseph B. Muskat & Stanley F. Robinson, "On the evaluation of Brewer's character sums," Trans. Amer. Math. Soc., v. 171, 1972, pp. 317347. MR 0306122 (46:5249)
 [5]
 Helmut Hasse, "Der te Potenzcharakter von 2 im Körper der ten Einheitswurzeln," Rend. Circ. Mat. Palermo (2), v. 7, 1958, pp. 185244. MR 0105401 (21:4143)
 [6]
 Richard H. Hudson & Kenneth S. Williams, A Class Number Formula for Certain Quartic Fields, Carleton Mathematical Series No. 174, Carleton University, Ottawa, 1981.
 [7]
 Richard H. Hudson, Kenneth S. Williams & Duncan A. Buell, "Extension of a theorem of Cauchy and Jacobi," J. Number Theory (To appear.) MR 769786 (86i:11002)
 [8]
 Richard H. Hudson & Kenneth S. Williams, "Binomial coefficients and Jacobi sums," Trans. Amer. Math. Soc., v. 281, 1984, pp. 431505. MR 722761 (85m:11092)
 [9]
 Emma Lehmer, "The quintic character of 2 and 3," Duke Math. J., v. 18, 1951, pp. 1118. MR 0040338 (12:677a)
 [10]
 Emma Lehmer, "On Euler's criterion," J. Austral. Math. Soc., v. 1, 1959, pp. 6470. MR 0108475 (21:7191)
 [11]
 C. R. Matthews, "Gauss sums and elliptic functions II. The quartic sums," Invent. Math., v. 54, 1979, pp. 2352. MR 549544 (81e:10035)
 [12]
 Joseph B. Muskat & YunCheng Zee, "On the uniqueness of solutions of certain Diophantine equations," Proc. Amer. Math. Soc., v. 49, 1975, pp. 1319. MR 0360461 (50:12911)
 [13]
 Bennett Setzer, "The determination of all imaginary, quartic, Abelian number fields with class number 1," Math. Comp., v. 35, 1980, pp. 13831386. MR 583516 (81k:12005)
 [14]
 Lothar Stickelberger, "Ueber eine Verallgemeinerung der Kreisteilung," Math. Ann., v. 37, 1890, pp. 321367. MR 1510649
 [15]
 Albert Leon Whiteman, "Theorems analogous to Jacobsthal's theorem," Duke Math. J., v. 16, 1949, pp. 619626. MR 0031940 (11:230b)
 [16]
 Koichi Yamamoto, "On a conjecture of Hasse concerning multiplicative relation of Gaussian sums," J. Combin. Theory, v. 1, 1966, pp. 476489. MR 0213311 (35:4175)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407582080
PII:
S 00255718(1984)07582080
Article copyright:
© Copyright 1984
American Mathematical Society
