Here are the ratios $[n(p)-1][n(p)-2]/P(p)$ (necessarily integers) for $p=3,5,\ldots,97$: 1, 1, 1, 19, 17, 1, 409, 604, 82, 20951, 229931, 411012, 39080794, 4382914408, 6345486566, 45119290746, 581075656330, 8672770990, 869561574799171, 71853663603175593, 25509154378676494, 24040267482771436703, 102403319155457392955, 11302410854347731819765

I mean, we also have $2\times3=P(3)$, $5\times6=P(5)$, $14\times 15=P(7)$, but are there any instances with $k\ge19$?

So $714\times 715=P(17)$. Is there another pair of consecutive integers so that $n(n+1)=P(k)$ for some $k$? Or is this impossible?

More generally, any biregular graph: en.wikipedia.org/wiki/Biregular_graph

Complete bipartite graph?

i.e. a ceiling.

What do you mean by "exactly the same argument?" The solution to a Sudoku puzzle is in no way the ground state of a normal computer.

The product of the first $k$ diagonal elements of $L$ squared is equal to the determinant of the $k\times k$ top left submatrix of $Q$.

@Alekk: Gram-Schmidt is a numerically unstable algorithm for producing a QR decomposition. It is better to used a canned QR decomposition routine from a numerical library.

@Elkies, I wonder if you saw this question from a couple of weeks ago, which also seems to be about intersection of cross polytopes with lattices: mathoverflow.net/questions/132165/…

see also mathoverflow.net/questions/132335/…

Of course it's not convex. Since if $e^y=x$ and $e^{y'}=x'$ then in general $e^{(y+y')/2}\neq (x+x')/2$. A more reasonable question would be if the its projection on $x$ or on $y$ is convex.

@Dima: no, consider the catenoid (approximated by a polyhedron).

This paper by Elkies, Odlyzko, and Rush gives a bound in the other direction, namely a lower bound for the packing density of $L_p$ balls: dx.doi.org/10.1007/BF01232282 . Hopefully Noam will stop by and tell us whether there is an easy upper bound.

@Douglas: the center point is needed. Consider the triangle with vertices at (-1/4,1/2), (-1/4,-1/4), and (1/2,-1/4). It has area 9/32 > 1/4.

@Frank, welcome to MO. The 2D wet foam was solved by L. Fejes Toth under the assumption of convex bubbles. It's possible that the way Hales got rid of the convexity assumption for dry foams might also work for wet foams.

I believe polymer people call the correlation time for this kind of dynamics the "Rouse relaxation time," so that should give a clue to search for how it is calculated.