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Concept

Asymptotic

A description of how a quantity behaves in the limit, ignoring everything that stays comparatively small. Two quantities are asymptotic when their ratio approaches one, which permits an error growing without bound so long as it grows more slowly.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Three properties that leave the middle, and one that cannot. Four measured curves of the share of random graphs having a property, plotted against the number of points: three first-order properties running to zero or one, and the parity of the edge count sitting on a half throughout.

Nearly always, or nearly never

Toss a coin for every pair of points and ask whether the graph that results has some property. For a property a first-order sentence can state, the answer in the limit is never a genuine probability — it is zero or it is one, and the game is what proves it.

logic · Ehrenfeucht–Fraïssé games Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes 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.

number · Sums of two squares The next sum of two squares after 150. A quarter of the circle of radius √150 on the integer lattice, with the column x = 12 and the first lattice point above the circle in it, (12, 3), on the circle of radius √153.

The wait for the next sum of two squares

Sums of two squares thin out to a share of nought, and yet the gaps between them stay short. Take the largest square below any number; what is left over is small, and a small number is always close to a square. Two squarings in a row say the next sum of two squares is never more than about 2√2 times the fourth root of n away — a bound proved in 1947 that nobody has improved, sitting far above every gap anyone has found.

number · Sums of two squares

Named alongside it

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

Exhaustive searchPrimesSums of two squaresApproximationConjectureConvergence rateCounting argumentDensityDivergenceElementary equivalenceExpressive powerHeuristic

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