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**(1962-63), 141-147. MR**26:2418****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.**P. Erdös,*Quelques problèmes de la théorie des nombres*, Monographies de l'Enseignement Math., No. 6 (Genève 1963), Problème 31. MR**28:2070****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.**B. Lindström,*An inequality for -sequences*, J. Comb. Theory**6**(1969), 211-212. MR**38:4436****6.**I. Z. Ruzsa,*Solving a linear equation in a set of integers*, Acta Arith.**65**(1993), 259-282. MR**94k:11112****7.**Z. Zhang,*Finding finite -sequences with larger*, Math. Comp.**63**(1994), 403-414. MR**94i:11109**

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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