Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

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 $\phi(x+k)=\phi(x)$, Acta Arith. 4 (1958), 181-184. MR 21:5597
  • 3. A. Schinzel and A. Wakulicz, Sur l'équation $\phi(x+k)=\phi(x)$. II, Acta Arith. 5 (1959), 425-426. MR 23:A831
  • 4. W. Sierpinski, Sur une propriété de la fonction $\phi(n)$, Publ. Math. Debrecen 4 (1956), 184-185. MR 18:17b

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11N25, 11Y99

Retrieve articles in all journals with MSC (2000): 11N25, 11Y99

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

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

American Mathematical Society