Concept

Bias — where it appears

A persistent lean in a quantity that is equal in the limit, showing up in a lower-order term rather than in the leading one. The primes leaving remainder 3 on division by 4 lead those leaving 1 at almost every bound, and the two classes still have the same density.

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

Which regions each rule favours. Average seats above or below exact quota for the largest and the smallest region, under each of the five methods, over 400 generated instances.

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

applied · Apportionment

The primes 3 mod 4 against the primes 1 mod 4, out to 30,000. A plot of the difference between the counts of primes in two residue classes, against the bound, showing a persistent lead for one class and where it is lost.

Every class, and in equal shares

Euclid's argument aimed at a residue class reaches some classes and stalls at others. The theorem covering all of them is Dirichlet's, its proof abandons arithmetic entirely for analysis, and what it proves is stronger than infinitude — the classes are equal, though not at any point anybody has counted.

number · Infinitude of primes

Named alongside it

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

Apportionment Arithmetic progression Density Dirichlet theorem Divisor method Expectation Geometric mean Modular arithmetic Monotonicity Prime Quota Rounding

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