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An upper bound on Jacobsthal's function


Authors: Fintan Costello and Paul Watts
Journal: Math. Comp. 84 (2015), 1389-1399
MSC (2010): Primary 11N25; Secondary 11Y55
DOI: https://doi.org/10.1090/S0025-5718-2014-02896-2
Published electronically: November 6, 2014
MathSciNet review: 3315513
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Abstract | References | Similar Articles | Additional Information

Abstract: The function $ h(k)$ represents the smallest number $ m$ such that every sequence of $ m$ consecutive integers contains an integer coprime to the first $ k$ primes. We give a new computational method for calculating strong upper bounds on $ h(k)$.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdős, On the integers relatively prime to $ n$ and on a number-theoretic function considered by Jacobsthal, Math. Scand. 10 (1962), 163-170. MR 0146125 (26 #3651)
  • [2] Thomas R. Hagedorn, Computation of Jacobsthal's function $ h(n)$ for $ n<50$, Math. Comp. 78 (2009), no. 266, 1073-1087. MR 2476571 (2009k:11146), https://doi.org/10.1090/S0025-5718-08-02166-2
  • [3] Lajos Hajdu and Natarajan Saradha, On a problem of Pillai and its generalizations, Acta Arith. 144 (2010), 323-347.
  • [4] L. Hajdu and N. Saradha, Disproof of a conjecture of Jacobsthal, Math. Comp. 81 (2012), no. 280, 2461-2471. MR 2945166, https://doi.org/10.1090/S0025-5718-2012-02581-6
  • [5] Henryk Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), no. 1, 225-231. MR 499895 (80h:10053)
  • [6] Hans-Joachim Kanold, Über eine zahlentheoretische Funktion von Jacobsthal, Math. Ann. 170 (1967), 314-326 (German). MR 0209247 (35 #149)
  • [7] Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Trans. Amer. Math. Soc. 322 (1990), no. 1, 201-237. MR 972703 (91b:11093), https://doi.org/10.2307/2001529
  • [8] János Pintz, Very large gaps between consecutive primes, J. Number Theory 63 (1997), no. 2, 286-301. MR 1443763 (98c:11092), https://doi.org/10.1006/jnth.1997.2081
  • [9] Carl Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory 12 (1980), no. 2, 218-223. MR 578815 (81m:10081), https://doi.org/10.1016/0022-314X(80)90056-6
  • [10] Harlan Stevens, On Jacobsthal's $ g(n)$-function, Math. Ann. 226 (1977), no. 1, 95-97. MR 0427212 (55 #247)
  • [11] The PARI Group, Bordeaux, PARI/GP, version 2.5.3, 2012, available from http://pari.math.u-bordeaux.fr/.

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Additional Information

Fintan Costello
Affiliation: School of Computer Science and Informatics, University College Dublin, Belfield, Dublin 6, Ireland
Email: fintan.costello@ucd.ie

Paul Watts
Affiliation: Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland
Email: watts@thphys.nuim.ie

DOI: https://doi.org/10.1090/S0025-5718-2014-02896-2
Received by editor(s): May 24, 2012
Received by editor(s) in revised form: September 16, 2013, and September 26, 2013
Published electronically: November 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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