Finding finite -sequences faster

Author:
Bernt Lindström

Journal:
Math. Comp. **67** (1998), 1173-1178

MSC (1991):
Primary 11B75, 11Y55, 12E20

DOI:
https://doi.org/10.1090/S0025-5718-98-00986-7

MathSciNet review:
1484901

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Abstract | References | Similar Articles | Additional Information

Abstract: A -sequence is a sequence of positive integers such that the sums , , are different. When is a power of a prime and is a primitive element in then there are -sequences of size with , which were discovered by R. C. Bose and S. Chowla.

In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence by integers relatively prime to the modulus is equivalent to varying . Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over when is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in ``Primitive quadratics reflected in -sequences'', to appear in *Portugaliae Mathematica* (1999).

**1.**R. C. Bose and S. Chowla,*Theorems in the additive theory of numbers*, Comment. Math. Helv.**37**(1962/1963), 141–147. MR**0144877**, https://doi.org/10.1007/BF02566968**2.**R. C. Bose, S. Chowla and C. R. Rao,*On the integral order of quadratics , with applications to the construction of minimum functions for and to some number theory results*, Bull. Calcutta Math. Soc.**36**(1944), 153-174. MR**6:256b****3.**Paul Erdős,*Quelques problèmes de théorie des nombres*, Monographies de L’Enseignement Mathématique, No. 6, L’Enseignement Mathématique, Université, Geneva, 1963, pp. 81–135 (French). MR**0158847****4.**P. Erdös and P. Turán,*On a problem in additive number theory*, J. London Math. Soc.**16**(1941), 212-215; ibid**19**(1944), 208.**5.**Bernt Lindström,*An inequality for 𝐵₂-sequences*, J. Combinatorial Theory**6**(1969), 211–212. MR**0236138****6.**Imre Z. Ruzsa,*Solving a linear equation in a set of integers. I*, Acta Arith.**65**(1993), no. 3, 259–282. MR**1254961****7.**Zhen Xiang Zhang,*Finding finite 𝐵₂-sequences with larger 𝑚-𝑎^{1/2}_{𝑚}*, Math. Comp.**63**(1994), no. 207, 403–414. MR**1223235**, https://doi.org/10.1090/S0025-5718-1994-1223235-2

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Additional Information

**Bernt Lindström**

Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

Address at time of publication:
Turbingränd 18, S-17675 Järfälla, Sweden

DOI:
https://doi.org/10.1090/S0025-5718-98-00986-7

Keywords:
$B_2$-sequence,
Bose-Chowla theorem,
finite field,
primitive element,
primitive quadratic

Received by editor(s):
November 21, 1996

Article copyright:
© Copyright 1998
American Mathematical Society