Vol. 69, No. 1, 1977

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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