Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Finding finite $B_ 2$-sequences with larger $m-a^ {1/2}_ m$


Author: Zhen Xiang Zhang
Journal: Math. Comp. 63 (1994), 403-414
MSC: Primary 11Y55; Secondary 11B75
DOI: https://doi.org/10.1090/S0025-5718-1994-1223235-2
MathSciNet review: 1223235
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A sequence of positive integers ${a_1} < {a_2} < \cdots < {a_m}$ is called a (finite) ${B_2}$-sequence, or a (finite) Sidon sequence, if the pairwise differences are all distinct. Let \[ K(m) = \max (m - a_m^{1/2}),\] where the maximum is taken over all m-element ${B_2}$-sequences. Erdős and Turán ask if $K(m) = O(1)$. In this paper we give an algorithm, based on the Bose-Chowla theorem on finite fields, for finding a lower bound of $K(p)$ and a p-element ${B_2}$-sequence with $p - a_p^{1/2}$ equal to this bound, taking $O({p^3}{\log ^2}pK(p))$ bit operations and requiring $O(p\log p)$ storage, where p is a prime. A search for lower bounds of $K(p)$ for $p \leq {p_{145}}$ is given, especially $K({p_{145}}) > 10.279$, where ${p_i}$ is the ith prime.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11Y55, 11B75

Retrieve articles in all journals with MSC: 11Y55, 11B75


Additional Information

Keywords: <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${B_2}$">-sequences, Erd&#337;s-Tur&#225;n conjecture, Bose-Chowla theorem, finite fields, algorithms
Article copyright: © Copyright 1994 American Mathematical Society