Concept

Conjecture

A statement believed true and not proved, usually on the strength of computation or of a pattern holding in every case anybody has checked. Its value is in what it organises, and the proof of a good one generally needs machinery invented for the purpose.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

discrete · prime distribution

Every fifth partition count divides, and the rank that says why. A row of partition counts with the ones in a congruence class marked, and a histogram of partitions sorted by rank.

Every fifth one divides

p(4) is 5, p(9) is 30, p(14) is 135, and every partition count at a number leaving four on division by five is divisible by five. Ramanujan read it off a table; the explanation is a way of splitting those partitions into five equal heaps.

number · partitions

A path of 4 bounces that closes, in a triangle of 100°, 40°, 40°. A triangular billiard table with a periodic path found by an exhaustive sweep of starting positions and directions.

The triangle nobody can settle

Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.

dynamics · billiards

Named alongside it

The objects these essays reach for when they reach for this one.

Billiards Classification Congruence Counterexample Counting argument Cramér's model Decision procedure Density Divisibility Equivalence Exhaustive search Generating function

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