The minimal number of solutions to $\phi (n)=\phi (n+k)$
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- by Jeffrey J. Holt;
- Math. Comp. 72 (2003), 2059-2061
- DOI: https://doi.org/10.1090/S0025-5718-03-01509-6
- Published electronically: February 3, 2003
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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
- L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math. 33 (1904), 155–161.
- A. Schinzel, Sur l’équation $\phi (x+k)=\phi (x)$, Acta Arith. 4 (1958), 181–184 (French). MR 106867, DOI 10.4064/aa-4-3-181-184
- A. Schinzel and Andrzej Wakulicz, Sur l’équation $\phi (x+k)=\phi (x)$. II, Acta Arith. 5 (1959), 425–426 (1959) (French). MR 123506, DOI 10.4064/aa-5-4-425-426
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Bibliographic 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
- 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.
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 2059-2061
- MSC (2000): Primary 11N25; Secondary 11Y99
- DOI: https://doi.org/10.1090/S0025-5718-03-01509-6
- MathSciNet review: 1986821