On the special cases of Carmichael’s totient conjecture

Euler's totient function, 𝜑(n), is the arithmetic function defined as the number of positive integers less than or equal to n that are relatively prime to n. In his 1922 paper [3], Professor R. D. Carmichael conjectured that for each positive integer n, there exists at least one positive integer m ≠ n such that 𝜑(m) = 𝜑(n). In this paper, we consider some relevant literature and explore Carmichael's totient conjecture for particular values of 𝜑(n) = k. Our main consideration will be the set X_k = {n ∈ ℕ : 𝜑(n) = k}. In identifying X_k for k = 2^t, 2p^s, 2^2p, and 2pq, where p and q are distinct prime numbers, we find that Carmichael's conjecture holds for those select cases, provide an algorithm, and some related results. The conjecture remains an open problem in number theory [9].