The Family Xqpˢ for Distinct Primes $q,p$ in Connection with Carmichael's Totient Conjecture

Shannon, Shiue, Saito and the author {SSHS2025} studied the sets Xₖ=:φ(n)=k\ in connection with Carmichael's totient conjecture, and proved that for an odd prime p≠3 and s∈\1,2,3,4\, X2pˢ is empty whenever 2pˢ+1 is not prime (with $s=1$ fully characterized in both directions), conjecturing that this persists for every s≥1. We resolve that conjecture in full, obtaining a complete characterization of X2pˢ for every s≥1 and every odd prime p≠3, together with a parity dichotomy: X2pˢ= unconditionally whenever s is even, while for odd s both outcomes genuinely occur, determined exactly by the primality of 2pˢ+1. We then close the one remaining gap at $p=3$, where the proof technique degenerates, with a separate, unconditional characterization of X2·3ˢ. Building on these results, we consider the natural generalization Xqpˢ, where q≠ p are distinct primes and s≥1. We show that this generalization splits into three cases of very different character. If q and p are both odd, the problem is trivial: qpˢ is odd, so Xqpˢ= by elementary parity, with no case-analysis needed. If $q=2$, the problem is exactly the family X2pˢ resolved above. If $p=2$, giving the ``mirror'' family X2ˢq for an odd prime q, the problem is genuinely different. Beyond $s=1$, we give a complete resolution for $s=2$, recovering, with an explicit proof and solution sets, the qualitative dichotomy of [Thm.~2.3.1]{SSHS2025}, but the problem remains open for s≥3.