On an arithmetic function considered by Pillai

Florian Luca[1]; Ravindranathan Thangadurai[2]

  • [1] Mathematical Institute UNAM, Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, Mexico
  • [2] Harish-Chandra Research Institute Chhatnag Road, Jhunsi Allahabad 211 019, India

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 695-701
  • ISSN: 1246-7405

Abstract

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For every positive integer n let p ( n ) be the largest prime number p ≤ n . Given a positive integer n = n 1 , we study the positive integer r = R ( n ) such that if we define recursively n i + 1 = n i - p ( n i ) for i ≥ 1 , then n r is a prime or 1 . We obtain upper bounds for R ( n ) as well as an estimate for the set of n whose R ( n ) takes on a fixed value k .

How to cite

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Luca, Florian, and Thangadurai, Ravindranathan. "On an arithmetic function considered by Pillai." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 695-701. <http://eudml.org/doc/10905>.

@article{Luca2009,
abstract = {For every positive integer $n$ let $p(n)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R(n)$ such that if we define recursively $n_\{i+1\}=n_i-p(n_i)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R(n)$ as well as an estimate for the set of $n$ whose $R(n)$ takes on a fixed value $k$.},
affiliation = {Mathematical Institute UNAM, Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, Mexico; Harish-Chandra Research Institute Chhatnag Road, Jhunsi Allahabad 211 019, India},
author = {Luca, Florian, Thangadurai, Ravindranathan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Pillai function; growth of arithmetic function},
language = {eng},
number = {3},
pages = {695-701},
publisher = {Université Bordeaux 1},
title = {On an arithmetic function considered by Pillai},
url = {http://eudml.org/doc/10905},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Luca, Florian
AU - Thangadurai, Ravindranathan
TI - On an arithmetic function considered by Pillai
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 695
EP - 701
AB - For every positive integer $n$ let $p(n)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R(n)$ such that if we define recursively $n_{i+1}=n_i-p(n_i)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R(n)$ as well as an estimate for the set of $n$ whose $R(n)$ takes on a fixed value $k$.
LA - eng
KW - Pillai function; growth of arithmetic function
UR - http://eudml.org/doc/10905
ER -

References

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  1. R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes - II. Proc. London Math. Soc., (3) 83 (2001), 532–562. Zbl1016.11037MR1851081
  2. H. Cramér, On the order of magnitude of the differences between consecutive prime numbers. Acta. Arith., 2 (1936), 396–403. Zbl0015.19702
  3. H. Halberstam and H. E. Rickert, Sieve methods. Academic Press, London, UK, 1974. Zbl0298.10026
  4. G.  Hoheisel, Primzahlprobleme in der Analysis.   Sitzunsberichte  der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33 (1930), 3–11. 
  5. T. R. Nicely, Some Results of Computational Research in Prime Numbers. http://www.trnicely.net/ 
  6. S.  S.  Pillai, An arithmetical function concerning primes. Annamalai University J. (1930), 159–167. 
  7. R. Sitaramachandra Rao, On an error term of Landau - II in “Number theory (Winnipeg, Man., 1983)”, Rocky Mountain J. Math. 15 (1985), 579–588. Zbl0584.10027MR823269

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