As we know, there are lots of consequences with the presupposition of the Riemann Hypothesis.
Similarly, are there any important consequences with the presupposition of $\mathbf{P} \neq \mathbf{NP}$ ?
An alternative statement of $\mathbf{P} \neq \mathbf{NP}$ is the extended Church-Turing Thesis. So if we have an speedup algorithm of other model than classic Turing Machine, we have to find an new algorithm for Turing Machine with the assumption $\mathbf{P} \neq \mathbf{NP}$ ? that means we have to find new speedup algorithm of factoring and the like.
