General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

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(x(logx)). We also prove that any such n with precisely K distinct prime factors is necessarily less than K. There are appropriate generalizations of these results to integers n for which φ{n) \ n — a, a an arbitrary integer.

/pdf/on-composite-n-for-which-ph-n-n-1-ii-1b3youvw23.pdf

On composite n for which φ(n)∣n − 1. II

/pdf/a-note-on-p-spaces-and-moore-spaces-1ia67ak136.pdf

A note on p-spaces and Moore spaces

/pdf/mean-value-iteration-of-nonexpansive-mappings-in-a-banach-1ffoen31r4.pdf

Mean value iteration of nonexpansive mappings in a Banach space

/pdf/geometric-theory-of-a-single-markov-operator-2rr9mvhlch.pdf

Geometric theory of a single Markov operator

/pdf/on-rings-with-proper-involution-39g5lzoh4z.pdf

On rings with proper involution

https://msp.org/pjm/1969/30-3/pjm-v30-n3-p01-s.pdf

Topological rings with property $(Y)$.

/pdf/rings-with-involution-and-the-prime-radical-3naobw3ss1.pdf

Rings with involution and the prime radical

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On rings with proper involution

Topological rings with property $(Y)$.

Rings with involution and the prime radical