Vol. 69, No. 1, 1977

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
On composite n for which φ(n)n 1. II

Carl Pomerance

Vol. 69 (1977), No. 1, 177–186

The problem of whether there exists a composite n for which φ(n)n 1 (φ is Euler’s function) was first posed by D. H. Lehmer in 1932 and still remains unsolved. In this paper we prove that the number of such n not exceeding x is O(x12(log x)34). We also prove that any such n with precisely K distinct prime factors is necessarily less than K2K . There are appropriate generalizations of these results to integers n for which φ(n)n a, a an arbitrary integer.

Mathematical Subject Classification
Primary: 10A20, 10A20
Received: 28 October 1975
Published: 1 March 1977
Carl Pomerance
Mathematics Department
Dartmouth College
Kemeny Hall
Hanover NH 03755
United States