This is a preview. Log in through your library . Abstract 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 ...

An old conjecture of Sierpiński asserts that for every integer k ≥ 2, there is a number m for which the equation φ (x) = m has exactly k solutions. Here φ is Euler's totient function. In 1961, ...