What is the state of Carmichael's totient function conjecture?

Question

I have been searching for information about that conjecture and it seems for me that noone has made any significant improvement on it in the last 30 years.

Is that true? Does it remain unproven to be true? Has there been any important discovery about the problem in the last years?

Thank you.

Answer 1

Let's define Carmichael's Totient Conjecture:

For each $n$, there exists an integer $m\neq n$ such that $φ(m) = φ(n) = k$. Where $φ$ defined to be Euler's Totient.

The conjecture is an open problem in general, but is proven for all $k$ such that $k+1$ is prime.

Proof: Suppose $n$ is prime and $n-1 = k$. Then $φ(n) = k$. Now $φ(2) = 1$, and the totient of any integer $t$ is the product of totients of primes powers dividing $t$. Now let $m = 2n$. Since the only prime powers dividing $m$ are $2$ and $n$, $φ(m) = (2-1)*(n-1)$ $=$ $φ(m) = k$, therefore $φ(m) = φ(n) = k$.