P ≠ NP

In 2000 were mooted seven Problems for a "Millennium Prize", to succeed (and in parts include) Hilbert's twenty-three.

I have mentioned the Continuum Hypothesis, which was one of Hilbert's; a new set theory is still outstanding, but the mathematicians have demoted that from the top six.

We are not counting Henri Poincaré in this six because his Conjecture is a Theorem now - it has been solved. The whole Geometrisation Conjecture has been solved; although mathematicians didn't buy Perelman's proof at the time (2003) they did iron it out eventually (2007). Any four-dimensional manifold must exist in any one of eight geometries. Given that our universe is an instance of one of those 4-D manifolds (probably), this matters to future studies of local physics. And to theologians, perhaps.

Of the six remaining Problems, perhaps the hottest Problem is P Or Not To P. To whit: whether a nonpolynomial problem, like an expansion into factorial or otherwise-exponential (that is, f(x)= 2^x) possibilities, may be solved by a polynomial-step method - by such a f(x)= x² "algorithm", as the Arabs call it. The traveling salesman's eternal question of how to get his route to multiple houses done shortest (or, fastest) is the classic instance of "NP complete".

I have had an interest in this since I first did Comp Sci in college. All programmers know P v NP; it is a matter of survival. If our boss demands a quick solution to some NP problem, once we have found out that it is in fact NP, we tell our boss LOLGF - we can't do it. 'Tis one of those problems. They shoot horses don't they?

I am not here to solve P v NP formally. I'm sorry, HAL; but I can't do that. I will, here, lay down my marker that P ≠ NP.

The Inductive Method is the algorithm which proves functions true, or not, within a given infinite-but-countable set. Georg Cantor found stuff which the inductive method doesn't find. The stuff which Cantor flagged as unreachable to an inductive algorithm is exactly an exponential function. f(x)= 2^x. Plus or minus four.

Algebraic numbers are countable; they are subject to induction. Those are the numbers accessible to polynomial functions. Transcendental numbers aren't so accessible; among which is the natural logarithm of "one". There is no polynomial function that can net you a transcendental number like e (or pi). I posit there is no polynomial-time algorithm that can solve a problem with e-to-the-power-that-time variables.

Again : I am not calling this highschool-level blog post OMGz PROOF P ≠ NP. I will, however, declare it as sufficient as intuition. Nobody is going to find a polynomial-time solution to a nonpolynomial problem in my lifetime. Or in yours.