Generator

reciprocity

A generator in the number library, called 4 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

reciprocity is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Counting a 5 by 3 rectangle two waysLattice points in a rectangle cut by a diagonal of slope q over p, coloured by which side they fall.slope 7/117 points below the line, 8 above, and 7 + 8 = 5 × 3 = 15(7 | 11) = −1 from the count below; (11 | 7) = +1 from the count above

show: "gauss"

The multiples of 5, modulo 13Each multiple marked on a strip of residues, with the ones in the top half folded down.01234567891011121 × 552 × 5103 × 524 × 575 × 5126 × 543 of the 6 multiples had to be folded down from the top halfso (5 | 13) = (−1)^3 = −1: 5 is not a square modulo 13

What it checks while it draws

Collected by running the family and listening to lib/verify.js, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

The whole library · What the figures prove