I prove that, if there is a solution x for some n > 1, then x - n
= y is strictly less than 4n^2.
Suppose y satisfies the equation
sigma(y) = phi(y + 2n) (**)
and suppose y >= 4n^2.
If y has a prime divisor p <= Sqrt(y) then it also has a
divisor d = y/d >= Sqrt(y).
Thus, sigma(y) > y+d >= y+Sqrt(y) >= y+2n, which contradicts the
equality (**), since phi(m) is allways less than m. Thus, y
is prime and sigma(y) = y+1.
From (**) it is clear that y+2n = z cannot be prime, unless n =
1. Thus, z has a prime divisor p <= sqrt(z). Since phi(z) =
z*Product(1 - 1/r), where r runs through all prime divisors of
z, it follows :
phi(z) <= z(1-1/p) = z - z/p <= z -
Sqrt(z) = y+2n - Sqrt(y+2n) < y+2n - Sqrt(y) <= y+2n - 2n = y.
This is a contradiction which shows that no solution y can be
bigger than 4n^2.
Could it also be worthwile to mention
that I found at least one solution for all n <= 10000 ?
Therefore, I think the conjecture that for every n > 2
there are finetely many solutions of (n,*) is very