Pet peeve of mine here. What you really mean is “no *USEFUL* closed form”. It is trivial to find closed form formulas for anything you can compute. In fact, I recently created a program which *finds* formulas for anything you can compute: http://www.xamuel.com/formula.php The formulas it produces are utterly useless in practice (and they’re so complicated they’re something of a crime against humanity), but that’s not the point.

]]>A — B — C

A — C — B

(in here letters stand for nodes)

]]>This is the slightly slicker way of thinking about the algorithm I mentioned in my comment.

]]>Suppose that p(x) = a_0 + a_1 x + a_2 x^2 + … + a_n x^n. Consider what you get when you divide p(m) by m. As m > a_0 + a_1 + … + a_n, you know that the term a_0/m must be less than 1. Therefore p(m)/m = (something less than one) + a_1 + m(a_2 + … + a_n m^(n-1) ). So if you know m and p(m) you can read off a_1 from this. Now consider p(m) / m^2 = (something less than one) + a_2 + m(something). Is it now clear how you can read off the a_i? It hinges on the fact that (a_0+…a_i)/m < 1 for any i between 0 and n, by the choice of m.

]]>And yes, I’m not claiming that P != NP has been settled; I only meant that intuitively it seems ludicrous that they might be equal (to me, at least).

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