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Concept

Elliptic curve

A smooth curve of degree three, such as y² = x³ + ax + b, whose points can be added: a line through two of them meets the curve at a third. That addition turns collinearity into arithmetic, and it governs counts of solutions modulo primes.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The cubic curves over GF(43) with the most and the fewest points. The solutions of two equations y squared equals x cubed plus ax plus b over the field with 43 elements, drawn as dots on a square grid: the curve with the most points and the curve with the fewest.

Give or take twice the square root

A cubic curve over the integers mod 43 should have about 44 points — one for each value of x, on average, and one at infinity. No curve misses by more than 13, the largest whole number below 2√43, and every count from 31 to 57 belongs to some curve. The first fact is Hasse's theorem, the second Deuring's, and the way the counts spread between the limits is a semicircle.

computation · Finite fields 9 trees on a cubic, 10 rows of three. The curve y = 1/(1 + x²) with 8 marked points on it and a 9th at infinity. Every line through three of them is drawn: 10 in all.

Rows of three, planted on a cubic

Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.

geometry · Ordinary lines Five triangles between the same two circles, every one closing. An outer circle of radius 1 and an inner of radius 0.38 at Euler's distance 0.4899; five triangles inscribed in the first and circumscribed about the second, started from different points.

A triangle that fits once fits everywhere

Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.

geometry · Triangle centres Right triangles of area five, six and seven. area 5: 3/2, 20/3, 41/6; area 6: 3, 4, 5; area 7: 24/5, 35/12, 337/60.

The areas a right triangle can have

A right triangle with sides 3, 4 and 5 has area 6; one with sides 3/2, 20/3 and 41/6 has area 5. No right triangle with rational sides has area 1, 2 or 3. Which whole numbers are areas is a thousand-year-old question, and its best answer — two counts of solutions to quadratic equations — rests on one of the great unproved conjectures.

geometry · Pythagoras The numbers below a hundred as sums of three cubes. Smallest solution size per n < 100: 1:9 2:1 3:1 6:2 7:2 8:16 9:2 10:2 11:3 12:11 15:2 16:2 17:2 18:3 19:3 20:3 21:16 24:2 25:3 26:3 27:6 28:3 29:3 30:10d 33:16d 34:3 35:3 36:3 37:4 38:4 39:6d 42:17d 43:3 44:8 45:4 46:3 47:8 48:4 51:796 52:11d 53:3 54:3 55:3 56:4 57:4 60:5 61:5 62:3 63:4 64:6 65:4 66:4 69:5 70:21 71:4 72:4 73:4 74:15d 75:9d 78:55 79:35 80:4 81:3 82:14 83:4 84:8d 87:4271 88:5 89:7 90:4 91:4 92:4 93:7 96:22 97:5 98:5 99:4.

Thirty-three as three cubes

Which numbers are a sum of three cubes, allowing negatives? Nothing like the rule for two squares is known — only that numbers leaving 4 or 5 on division by 9 are impossible. Every other number below a hundred has a sum with cubes under 5,000, except eight; 33 needed sixteen-digit cubes and was found in 2019.

number · Sums of two squares

Named alongside it

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

ConjectureCounting argumentExhaustive searchModular arithmeticCircumcircleCongruent numberCyclic groupDiophantine equationEuler relationExtremal configurationFinite fieldGenus

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