From: prezky@apple.com (Michael Press)
Subject: Euler's totient function
Date: Thu, 12 Oct 2000 16:12:16 -0700
Newsgroups: sci.math
Summary: [missing]

Paulo Ribenboim's _The Book of Prime Number Records_ is a treat. 
Today I was dazzled by the section on Euler's totient function,
and repeat here a few results that particularly captured my
imagination.

For every k >= 1, 2.7^k is not a value of Euler's totient function.  
Schnizel [1956]

There exist infinitely may primes p such that for every k>=1,
p.2^k is not a value of Euler's totient function.
Mendelsohn [1976]

For every k>=1, there exists n such that phi(n) = k!.  
Gupta [1950]

This next is wild.  

            phi(n+1)
lim sup ______________  = oo
 n->oo      phi(n)
 
            phi(n+1)
lim inf ______________  = 0
 n->oo      phi(n)
 
Somayajulu [1950]

The set of all numbers phi(n+1)/phi(n) is dense in the set of all
positive real numbers.  

The set of all numbers phi(n+1)/n is dense in (0,1).  

Schnizel, A.        [1956]  Sur l'equation phi(x) = m.  
                    Elem. Math., 11,1956, 75-78.  
Mendelsohn, N.S.    [1976]  The equation phi(x) = k.  
                    Math. Mag., 49, 1976, 37-39.  
Gupta, H.           [1950]  On a problem of Erdos.  
                    Amer. Math. Montly, 57, 1950, 326-329.  
Somayajulu, B.S.K.R [1950]  On Euler's totient function phi(n).  
                    Math. Student, 18, 1950, 31-32.

-- 
Michael Press