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The minimal number of solutions to $\phi(n)=\phi(n+k)$


Author: Jeffrey J. Holt
Journal: Math. Comp. 72 (2003), 2059-2061
MSC (2000): Primary 11N25; Secondary 11Y99
Published electronically: February 3, 2003
MathSciNet review: 1986821
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1958, A. Schinzel showed that for each fixed $k\leq 8\cdot 10^{47}$there are at least two solutions to $\phi(n)=\phi(n+k)$. Using the same method and a computer search, Schinzel and A. Wakulicz extended the bound to all $k \leq 2\cdot 10^{58}$. Here we show that Schinzel's method can be used to further extend the bound when $k$ is even, but not when $k$ is odd.


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

  • 1. L. E. Dickson, A new extension of Dirichlet's theorem on prime numbers, Messenger of Math. 33 (1904), 155-161.
  • 2. A. Schinzel, Sur l’équation 𝜑(𝑥+𝑘)=𝜑(𝑥), Acta Arith 4 (1958), 181–184 (French). MR 0106867
  • 3. A. Schinzel and Andrzej Wakulicz, Sur l’équation 𝜑(𝑥+𝑘)=𝜑(𝑥). II, Acta Arith. 5 (1959), 425–426 (1959) (French). MR 0123506
  • 4. W. Sierpiński, Sur une propriété de la fonction 𝜑(𝑛), Publ. Math. Debrecen 4 (1956), 184–185 (French). MR 0079023

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Additional Information

Jeffrey J. Holt
Affiliation: Department of Mathematics, Randolph-Macon College, Ashland, Virginia 23005
Address at time of publication: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: jjholt@virginia.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01509-6
Received by editor(s): August 14, 1998
Received by editor(s) in revised form: March 5, 2002
Published electronically: February 3, 2003
Additional Notes: The author was partially supported by a grant from the Walter Williams Craigie Endowment.
Article copyright: © Copyright 2003 American Mathematical Society