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Mathematics of Computation

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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
MathSciNet review: 1223235
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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.

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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