Concept

Quadratic residue

A number that is the square of something else in modular arithmetic. Exactly half the non-zero residues modulo an odd prime are squares, and joining two points when their difference is one gives colourings with unusually few large single-coloured sets.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Counting a 5 by 3 rectangle two ways. Lattice points in a rectangle cut by a diagonal of slope q over p, coloured by which side they fall.

Counting one rectangle, twice

Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.

number · quadratic reciprocity

17 points coloured by whether their difference is a square. 17 points on a circle with every pair joined, coloured by whether the difference of their labels is a square modulo 17; the largest set of points all joined by one colour has 3 members.

Eighteen people, and the seventeen that escape

Among any eighteen people, four are mutual acquaintances or four are mutual strangers. Seventeen can be arranged so that neither happens, and the arrangement is not a lucky find — it is a rule about squares.

discrete · ramsey theory

Named alongside it

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

Modular arithmetic Complete graph Counting argument Counting two ways Existence proof Gauss lemma Graph Induction Lattice Legendre symbol Paley graph Parity

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