This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then
It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient.
The first few noncototients are The cototient of n are Least k such that the cototient of k is n are (start with n = 0, 0 if no such k exists) Greatest k such that the cototient of k is n are (start with n = 0, 0 if no such k exists) Number of ks such that k − φ(k) is n are (start with n = 0) Erdős (1913–1996) and Sierpinski (1882–1969) asked whether there exist infinitely many noncototients.
This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family
Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).