Conjecture

Prompt

Prove or disprove this conjecture extensively and comprehensively and answer in extremely, if couldn't prove then determine what current mathematics do not has that could prove this conjecture, so answer in extreme detail: Conjecture Let \(n\ge2\) be a natural number. Define: \(M(n)\): the smallest prime factor of \(n\); \(F(n)\): the largest prime factor of \(n\); \(\Omega(n)\): the total number of prime factors of \(n\), counted with multiplicity; \(\pi(n)\): the number of primes less than or equal to \(n\). Question: If the four quantities \[ \boxed{M(n),\ F(n),\ \Omega(n),\ \pi(n)} \] are known, is \(n\) uniquely determined? Equivalently, is the map \[ \boxed{ n\longmapsto \bigl(M(n),F(n),\Omega(n),\pi(n)\bigr) } \] injective on the natural numbers \(n\ge2\)? That is, do \[ M(n_1)=M(n_2), \] \[ F(n_1)=F(n_2), \] \[ \Omega(n_1)=\Omega(n_2), \] and \[ \pi(n_1)=\pi(n_2) \] necessarily imply \[ \boxed{n_1=n_2}? \] Conjecture: Yes. Knowing \(M(n),F(n),\Omega(n)\), and \(\pi(n)\) uniquely determines \(n\).