The same thing twice — page 17
Logic
What a proof about numbers can compute
The proofs of (o → o) → o → o without detours are the whole numbers, one for each count of how often the assumption is used. A proof of the implication from that formula to itself therefore takes a number and returns one, and removing its detours computes the answer. Enumerating all 14,659 such proofs up to size 16 finds 299 different functions — sums, products, squares, cubes, tests on zero — and not one of them subtracts. The predecessor, n − 1, is not among them, and a single line about polynomials says it never will be.
Number
The same bell on thinner sets
A typical whole number has about log log N prime factors, spread in a bell curve. That was a theorem about all numbers, built on the picture of each prime dividing independently. The numbers one less than a prime, and the numbers n² + 1, are far too thin for that picture to have any obvious claim on them, and on both the same bell appears — with means shifted by constants that come straight from how often each small prime divides them. The squares plus one that are prime, the case of a single factor, number 102,205 up to n = two million against 102,302 predicted; whether there are infinitely many is open.
Topology
Torus knots that fill the three-sphere
Hopf filled the three-sphere with circles, every two linked once. Let the circles turn at two different speeds instead — p turns one way while they make q the other — and the three-sphere is filled with (p, q) torus knots, trefoils when p and q are 2 and 3, every two of them linked exactly pq times. Two exceptional fibres remain plain circles; every knot links them q and p times, and they link each other once. Computed from the drawn curves for ten fibrations, every pair comes out at pq to six decimal places.