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Disproof of a conjecture of Jacobsthal

Authors: L. Hajdu and N. Saradha
Journal: Math. Comp. 81 (2012), 2461-2471
MSC (2010): Primary 11N64, 11Y55
Published electronically: March 26, 2012
MathSciNet review: 2945166
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Abstract: For any integer $ n\geq 1$, let $ j(n)$ denote the Jacobsthal function, and $ \omega (n)$ the number of distinct prime divisors of $ n$. In 1962 Jacobsthal conjectured that for any integer $ r\geq 1$, the maximal value of $ j(n)$ when $ n$ varies over $ {\mathbb{N}}$ with $ \omega (n)=r$ is attained when $ n$ is the product of the first $ r$ primes. We show that this is true for $ r\leq 23$ and fails at $ r=24$, thus disproving Jacobsthal's conjecture.

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Additional Information

L. Hajdu
Affiliation: University of Debrecen, Institute of Mathematics, and the Number Theory Research Group of the Hungarian Academy of Sciences, P.O. Box 12., H-4010 Debrecen, Hungary

N. Saradha
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Dr. Homibhabha Road, Colaba, Mumbai, India

Keywords: Jacobsthal function, coverings by primes
Received by editor(s): November 22, 2010
Received by editor(s) in revised form: June 20, 2011
Published electronically: March 26, 2012
Additional Notes: This research was supported in part by the OTKA grants K67580 and K75566, and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, cofinanced by the European Social Fund and the European Regional Development Fund.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.