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), 603-613

MSC:
Primary 11D09; Secondary 11E20

MathSciNet review:
758208

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

*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

*x*in the above partition of , in the complicated case that arises when the class number of

*K*is and .

**[1]**Duncan A. Buell and Richard H. Hudson,*Solutions of certain quaternary quadratic systems*, Pacific J. Math.**114**(1984), no. 1, 23–45. MR**755481****[2]**L. E. Dickson,*Cyclotomy and trinomial congruences*, Trans. Amer. Math. Soc.**37**(1935), no. 3, 363–380. MR**1501791**, 10.1090/S0002-9947-1935-1501791-3**[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, and Stanley F. Robinson,*On the evaluation of Brewer’s character sums*, Trans. Amer. Math. Soc.**171**(1972), 317–347. MR**0306122**, 10.1090/S0002-9947-1972-0306122-5**[5]**Helmut Hasse,*Der 2ⁿ-te Potenzcharakter von 2 im Körper der 2ⁿ-ten Einheitswurzeln*, Rend. Circ. Mat. Palermo (2)**7**(1958), 185–244 (German). MR**0105401****[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]**Duncan A. Buell, Richard H. Hudson, and Kenneth S. Williams,*Extension of a theorem of Cauchy and Jacobi*, J. Number Theory**19**(1984), no. 3, 309–340. MR**769786**, 10.1016/0022-314X(84)90075-1**[8]**Richard H. Hudson and Kenneth S. Williams,*Binomial coefficients and Jacobi sums*, Trans. Amer. Math. Soc.**281**(1984), no. 2, 431–505. MR**722761**, 10.1090/S0002-9947-1984-0722761-X**[9]**Emma Lehmer,*The quintic character of 2 and 3*, Duke Math. J.**18**(1951), 11–18. MR**0040338****[10]**Emma Lehmer,*On Euler’s criterion*. part 1, J. Austral. Math. Soc.**1**(1959/1961), no. part 1, 64–70. MR**0108475****[11]**C. R. Matthews,*Gauss sums and elliptic functions. II. The quartic sum*, Invent. Math.**54**(1979), no. 1, 23–52. MR**549544**, 10.1007/BF01391175**[12]**Joseph B. Muskat and Yun Cheng Zee,*On the uniqueness of solutions of certain Diophantine equations*, Proc. Amer. Math. Soc.**49**(1975), 13–19. MR**0360461**, 10.1090/S0002-9939-1975-0360461-9**[13]**Bennett Setzer,*The determination of all imaginary, quartic, abelian number fields with class number 1*, Math. Comp.**35**(1980), no. 152, 1383–1386. MR**583516**, 10.1090/S0025-5718-1980-0583516-2**[14]**L. Stickelberger,*Ueber eine Verallgemeinerung der Kreistheilung*, Math. Ann.**37**(1890), no. 3, 321–367 (German). MR**1510649**, 10.1007/BF01721360**[15]**Albert Leon Whiteman,*Theorems analogous to Jacobstahl’s theorem*, Duke Math. J.**16**(1949), 619–626. MR**0031940****[16]**Koichi Yamamoto,*On a conjecture of Hasse concerning multiplicative relations of Gaussian sums*, J. Combinatorial Theory**1**(1966), 476–489. MR**0213311**

Retrieve articles in *Mathematics of Computation*
with MSC:
11D09,
11E20

Retrieve articles in all journals with MSC: 11D09, 11E20

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1984-0758208-0

Article copyright:
© Copyright 1984
American Mathematical Society