Concept

Egyptian fraction

A sum of distinct unit fractions, the way scribes of ancient Egypt wrote every fraction. Which sums are shortest, and how large their denominators must be, are old questions with many open cases, such as the Erdős–Straus conjecture for 4/n.

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

Twelve's divisors make every total up to twenty-eight. Stacked bars for each total from 1 to 28, each built from distinct divisors of 12.

Divisors that make every amount

Twelve's divisors — 1, 2, 3, 4, 6, 12 — can be chosen to add to every whole number from one to twenty-eight. Numbers like that are called practical, a test on their primes decides them, and they turn out to be counted like the primes: about 1.336 x / log x of them below x. For practical numbers Goldbach's conjecture and the twin conjecture are theorems.

number · Perfect numbers

Filling a unit with the largest piece that fits. A unit bar filled greedily by 1/2, 1/3, 1/7, 1/43 and 1/1807, and below it what is left of 1 after each number of terms, from 1/2 down to about 10⁻⁵² after eight: 1/1806 after four.

The largest piece that still fits

Fill a unit with unit fractions, always taking the largest one that still fits: 1/2, 1/3, 1/7, 1/43, 1/1807. The denominators are Sylvester's sequence, each one more than the product of all before it, and after k of them what is left is exactly one over the next term minus one. Curtiss proved in 1922 that no k unit fractions get closer to 1 from below; an exact search confirms it to six terms. Applied to almost any other fraction, the same greedy rule is wasteful — 8/97 ends with a denominator of 150 digits.

number · Stern brocot

Named alongside it

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

Exhaustive search Asymptotics Density Divisor sum Goldbach Greedy algorithm Perfect number Prime number theorem Primes Recurrence Subset-sum Unit fraction

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