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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Finding finite $B_ 2$-sequences with larger $m-a^ {1/2}_ m$
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by Zhen Xiang Zhang PDF
Math. Comp. 63 (1994), 403-414 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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