Concept

Unit fraction

A fraction with numerator one, such as 1/7. Every positive fraction is a sum of distinct unit fractions, and the greedy rule — take the largest that fits — always finds such a sum, though not always a short one.

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

$The tree writes 4/17 as four unit fractions, and a search finds three. Left-neighbour chain 4/17, 3/13, 2/9, 1/5, 0/1, gaps 1/221, 1/117, 1/45, 1/5; three-term decomposition 1/5 + 1/30 + 1/510.$

Three unit fractions for every four over n

Two neighbouring fractions in the tree always differ by a unit fraction, so stepping down the tree writes any fraction as a sum of them — four over n in at most four steps. Erdős and Straus asked in 1948 whether three always suffice. Three identities settle every n except those leaving remainder 1 on division by 24, a search settles every one anyone has tried, and nobody has a proof.

number · Stern brocot

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.

Greedy algorithm Conjecture Egyptian fraction Exhaustive search Farey sequence Mediant Recurrence Residue Stern brocot tree

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