The play "Arcadia" features a precocious young woman discovering fractal geometry (and the second law of thermodynamics in an amazing two-fer.) A contemporary mathematician explains that actually drawing a strange attractor by hand would require years and years of effort. Here's a sample fractal we can use for comparison, the Barnsley Fern: Read more... )

We introduce a finite category Pup. A standard representation is the objects {inside, outside} and in addition to the identities there are two morphisms that form a bijection:

door: inside -> outside
door -1: outside -> insideThe literature conventionally refers to these operations as "let the dog out" and "let the dog in", respectively ( Read more... )

The generalized Fermat equation Ax^a + By^b = Cz^c has many integer solutions if 1/a + 1/b + 1/c > 1. It has rather fewer in the opposite case. I collected a bunch of references in this Quora question. The case a = 3, b = 2, c= 7 has been completely solved and there are a few "large" solutions including 9262^3+15312283^2=113^7 ( Read more... )

Using the technique described in my previous post, I attempted to answer this question on Quora: "What is the smallest possible difference between the sum of two perfect nth powers and a different nth power?" http://qr.ae/RO86UXRead more... )

In 1991 Daniel Bernstein published a short paper on a computational technique for solving diophantine equations of the form p(a) + q(b) = r(c) + s(d): https://cr.yp.to/papers/sortedsums.pdf He describes how a heap can be used to visit all sums p(a) + q(b), with a,b < H, in ( Read more... )

What is Pi? Although the typical definition in terms of circles is intuitive, it's not particularly well suited to analysis ( Read more... )

A blog entry on "Open Problems That Might Be Easy" mentions this one: Given a DFA, does it accept a string that is the binary representation of a prime number--- is this decidable?
I think this is sort of a tricky question in that it sounds simple but actually asks for very deep insights. To solve it you might have to know something about prime ( Read more... )

Here's a Bill Gross talk at TED about why startups fail. Mr. Gross identifies five factors that might influence a startup's success, analyzes 250 startups according to those factors, and analyzes the results ( Read more... )

1. Is there such thing as an uncomputable integer? Or, even if a function F(x) is uncomputable, does, say F(10^100) still count as computable because, hey, some Turing machine out there somewhere could output it! Or does a number count as computable only if we (humans) can distinguish that machine from some other machine that also outputs a big ( Read more... )

I answered this question about the Master Theorem (which is used to calculate the asymptotic behavior of divide-and-conquer algorithms): Why is the recurrence: T (n) = sqrt(2) *T (n/2) + log(n) a case 1 of master method?The source of confusion (see the comments to my answer) seemed to be that the original poster did not really understand the ( Read more... )