Concept

Quadratic field

The numbers a + b√d with a and b rational, for a fixed whole number d that is not a square. Their whole numbers form a ring in which factorisation may or may not be unique, and arguments there settle equations about ordinary integers.

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

The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region.

Counting the classes that break factorisation

The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

number · Unique factorisation $Five powers of two that are seven more than a square. How close 2ⁿ − 7 comes to a square for n from 3 to 40, as a fraction of the gap between neighbouring squares: squares at n = 3, 4, 5, 7, 15 (1, 3, 5, 11, 181 squared), and at no other n up to 10000.$

Five powers of two, seven more than a square

2³ − 7 = 1², 2⁴ − 7 = 3², 2⁵ − 7 = 5², 2⁷ − 7 = 11², 2¹⁵ − 7 = 181². Ramanujan asked in 1913 whether there are any more. Testing every exponent up to 10,000 finds none, and Nagell proved in 1948 that none exist, by moving to a number system where 2 splits into two primes and factorisation is still unique. The same five solutions decide when a two-error-correcting code can be perfect.

number · Unique factorisation

Named alongside it

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

Unique factorisation Class group Class number Congruence Diophantine equation Discriminant Exhaustive search Ideal Modular group Perfect code Quadratic form

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