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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

   

 

An old conjecture of Erdos-Turán on additive bases


Authors: Peter Borwein, Stephen Choi and Frank Chu
Journal: Math. Comp. 75 (2006), 475-484
MSC (2000): Primary 11B83, 05B20; Secondary 94A11, 68R05
Published electronically: September 9, 2005
MathSciNet review: 2176410
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Abstract | References | Similar Articles | Additional Information

Abstract: There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that $0=\delta_0<\delta_1<\delta_2<\delta_3\cdots$ is an increasing sequence of integers and

\begin{displaymath}s(z) : = \sum_{i=0}^\infty z^{\delta_i}. \end{displaymath}

Suppose that

\begin{displaymath}s^2(z) := \sum_{i=0}^\infty b_i z^i. \end{displaymath}

If $b_i>0$ for all $i$, then $\{b_n\}$ is unbounded.


Our main purpose is to show that the sequence $\{b_n\}$ cannot be bounded by $7$. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Stephen Choi
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: kkchoi@cecm.sfu.ca

Frank Chu
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pmc@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-05-01777-1
PII: S 0025-5718(05)01777-1
Keywords: Erd\H{o}s and Tur\'{a}n conjecture, additive basis
Received by editor(s): September 28, 2004
Received by editor(s) in revised form: November 15, 2004
Published electronically: September 9, 2005
Additional Notes: This research was supported in part by grants from NSERC of Canada and MITACS
The third author was supported by the NSERC Undergraduate Student Research Award.
Article copyright: © Copyright 2005 by the authors