Small cases lie — page 11
Number
A quarter of the bases at most
Carmichael numbers pass Fermat's test for every base they share no factor with, so Fermat's test can be fooled for ever. Watch the squarings on the way to aⁿ⁻¹ instead, and a prime allows only one pattern; 561 lets just 10 of its 320 bases through, and Monier and Rabin proved that no odd composite but 9 lets through more than a quarter. Four fixed bases prove every number below 3.2 billion prime or not.
Number
Most numbers are missing from the table
An N × N multiplication table has N² cells, and every product in it is less than or equal to N². Yet only 22.5% of the numbers that small turn up in the 10,000 × 10,000 table, and the share keeps falling. Erdős proved in 1955 that it falls to nought; Kevin Ford found in 2008 how slowly — like a tiny power of the logarithm, 0.0861 — because a product of two numbers has too many prime factors to be typical.
Topology
Curves that cross a line and close
Draw a closed curve that crosses a straight line 2n times and never crosses itself. Above the line it is a set of nested arches, below it another, and every such curve is a pair of the Catalan number's arch systems that happen to close into one loop. There are 1, 2, 8, 42, 262, 1,828 of them, and every count computed so far grows by a factor creeping towards 12.26 — a number nobody can prove, with an exponent conjectured from the physics of random surfaces.