Disproof of a conjecture of Jacobsthal

Authors:
L. Hajdu and N. Saradha

Journal:
Math. Comp. **81** (2012), 2461-2471

MSC (2010):
Primary 11N64, 11Y55

DOI:
https://doi.org/10.1090/S0025-5718-2012-02581-6

Published electronically:
March 26, 2012

MathSciNet review:
2945166

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Abstract | References | Similar Articles | Additional Information

Abstract: For any integer , let denote the Jacobsthal function, and the number of distinct prime divisors of . In 1962 Jacobsthal conjectured that for any integer , the maximal value of when varies over with is attained when is the product of the first primes. We show that this is true for and fails at , 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

Email:
hajdul@science.unideb.hu

**N. Saradha**

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

Email:
saradha@math.tifr.res.in

DOI:
https://doi.org/10.1090/S0025-5718-2012-02581-6

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.