Carmichael's conjecture states that if φ (x) = n, then φ (y) = n for some y ≠ x (φ is Euler's totient function). We show that the conjecture is valid for all x under 10 10,900,000. The main new idea ...

Let ø denote Euler's phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n ⩽ x such that q ∤ ø (n). Part of the ...