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Sums of two squares — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

    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.

    part 1 · number
  2. The two squares of 97, produced by division. A table of the division chain on 97 and a square root of minus one modulo it, with each row's quotient and remainder, the point at which the remainder falls below the square root marked, and the two squares that add to 97.

    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.

    part 2 · number
  3. How rare a sum of two squares is. Two curves against the logarithm of the bound: the fraction of numbers below it that are sums of two squares, falling; and that count times the square root of the logarithm, divided by the bound, which is nearly constant.

    Almost no number is one

    Sums of two squares look common — a sixth of all numbers up to a million are one. The fraction is falling to nothing, at a rate so slow that no computation will ever make it obvious, and the constant in front of it has been computed to fifty places and identified with nothing.

    part 3 · number

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