Norm — where it appears
Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.
One way to factor, and no other
Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.
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.
Circles that are diamonds and squares
The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.
The identity that multiplies sums of squares
A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.
The integers among the quaternions
The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.
What is lost at eight
Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.
The square that cannot be negative
Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.
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.
Two families of solutions, and a box that holds both
Replace the 1 in Pell's equation by 7 and x² − 2y² = 7 still has infinitely many solutions — but they fall into exactly two families, each one an orbit of the same multiplication, and every family has a member inside a box whose size is fixed in advance. How many families there are is then a count of factors, and 3 has none.
The polygon a lattice becomes from far away
Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.
A fraction on the circle forces a whole point
If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.
Units that form a lattice
In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.
How near nought a sum of roots can come
Five arrows of length one, each pointing at a 122nd root of unity, laid end to end, return to within fifteen millionths of where they started. No five of the 123rd roots come within a thousandth. Two, three and four arrows shrink their smallest miss in a tidy power of n; five scatter over four powers of ten from one n to the next, and nobody knows the law they follow.
How close two sums of square roots come
Two different sums of square roots of whole numbers are never equal: square roots of different squarefree numbers are independent, so no cancellation can make them agree. But they can come very close. Among sums of two square roots of numbers up to 320, √88 + √289 and √171 + √177 agree to eight decimal places. How many digits must be computed, in the worst case, to tell two such sums apart is not known — the best proof allows a number of digits doubly exponential in the length of the sums, the examples suggest only a modest polynomial, and the gap between them is why nobody knows whether the shortest route through points in the plane can be checked quickly.
A prime leaves no room for a comb
Take the table of powers of a root of unity, pick any rows and any equal number of columns, and take the determinant. At order 12 about one such determinant in six is exactly nought. At order 11, among 184,756 of them, none is, and in 1926 Chebotarev proved that none ever is at a prime. It means a signal on p places and its transform together occupy at least p + 1, while a comb on twelve places and its transform take 4 and 3.
Named alongside it
The objects these essays reach for when they reach for this one.
LatticeAlgebraic integerSums of two squaresUnique factorisationDescentEuclidean algorithmGaussian integersPrimesQuaternionCounting two waysDeterminantDiscriminant