Gaussian integers
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
Two squares, and a lattice
Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.
The two supplements, and where the eight comes from
The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.
Which primes a form takes
A prime is the sum of two squares exactly when it is 1 modulo 4. Change the form slightly, to x² + 27y², and no congruence on p decides it at all — which is where the elementary subject ends and its successor begins.
On the circle and never home
Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.
The two squares actually produced
Three proofs say a prime one more than a multiple of four is a sum of two squares, and not one of them hands over the squares. Running the Euclidean algorithm half-way does — and where to stop is the whole of the correctness argument.
Factoring uniquely with no way to divide
Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.
Which way a prime's two squares point
A prime one more than a multiple of four is a sum of two squares in exactly one way, and the two squares make a point on a circle. Draw that point for every such prime and ask which way it faces. It faces every way equally — the angles spread evenly over the sector, which Hecke proved by giving each angle a remainder and copying Dirichlet. Measured over seventy-four thousand primes, they are even more evenly spread than random angles would be.
![Primes in five thousand thin sectors. 581517 primes in (2e7, 4e7] in 4967 sectors; mean 117.08, variance ratio 0.874, 2 empty.](/f/primes-in-five-thousand-thin-sectors-90657fc7.svg)
Sectors that shrink with the primes
Hecke proved that the angles of the primes a² + b² are evenly spread, so every fixed sector eventually gets its share. Let the sector shrink as the primes grow, to width X^(−α), and the question is open beyond small α. Measured on the 581,517 such primes between twenty and forty million, wide shrinking sectors fluctuate far less than chance and thin ones as much as chance — and the thin sectors that go empty are not scattered: they gather beside the directions of small rational slope, most of all slopes like 1/3 and 1/1 whose two terms are both odd.
A walk on Gaussian primes stopped by a moat
Start at 1 + i and step from Gaussian prime to Gaussian prime, never more than 2 at a time: the walk reaches 720 primes and stops at 17 + 42i, surrounded by a moat. Allow steps of √10 and it reaches a quarter of a million primes before stopping at distance 1,024; steps of 4 carry it to 4,313. Whether some fixed step lets a walker reach infinity is a question Basil Gordon asked in 1962, and it is open.
Named alongside it
The objects these essays reach for when they reach for this one.
Sums of two squaresLatticePrimesModular arithmeticUnique factorisationNormAlgebraic integerCounting two waysDescentEquidistributionEuclidean algorithmLegendre symbol