Short Egyptian fractions
About this app
Every fraction a/b as a sum of distinct unit fractions: the greedy method against the shortest expansion, as a pie and a zoom cascade down to denominators with 25 digits. N(b) for all b ≤ 200 computed exactly, plus Erdős–Straus (4/n). Result from openai/math (family 025, formalised in Lean): the number of terms needed grows only like log log b, and that is best possible – Erdős problem 304.
What it is about
Ancient Egyptians wrote fractions as sums of distinct unit fractions 1/n, as in the Rhind papyrus. How many unit fractions does a/b need at most? The maximum over all numerators is called N(b). The greedy method of Fibonacci and Sylvester always terminates, but it can get long and produce huge denominators. In 1950 Erdős showed N(b) ≲ log b / log log b and conjectured the order log log b; Vose reached √log b in 1985. An AI-generated preprint in github.com/openai/math (family 025) claims that N(b) grows exactly like log log b, up to unspecified constants. This is Erdős problem no. 304. The catalogue lists the main theorem as formalised in Lean; the preprint is not peer-reviewed.
What the app shows
A fraction a/b is shown as a pie: on the left the greedy method lays in the largest fitting unit fraction piece by piece, on the right is a shortest decomposition. A magnifier zooms into the remaining gap, down to denominators with 25 digits. For all b ≤ 200 the decompositions and N(b) are computed exactly by exhaustive search. A chart shows the growth shapes log b / log log b, √log b and log log b. A separate panel covers the open Erdős–Straus question about 4/n.
Try it yourself
Load the example 5/121 and compare: the greedy method needs more terms there, with huge denominators, than the shortest decomposition. Pick a denominator and show its hardest numerator, the fraction that needs the most terms for that b. The growth panel shows how slowly these numbers rise. The theorem is about very large b; the curve only shows the shape of growth, and the proof does not build shortest decompositions, it shows that short ones exist.
Frequently asked questions
- What is an Egyptian fraction?
- An Egyptian fraction writes a fraction as a sum of distinct unit fractions, fractions with numerator 1. An example is 3/4 = 1/2 + 1/4. Every positive rational number can be written this way, usually in many different ways.
- How does the greedy algorithm for Egyptian fractions work?
- The greedy method of Fibonacci and Sylvester picks, at each step, the largest unit fraction that still fits into the remainder and subtracts it. It always stops after finitely many steps. It often produces more terms and much larger denominators than necessary.
- How many unit fractions does a/b need at most?
- Erdős conjectured in 1950 that N(b) grows like log log b. An AI-generated preprint in the catalogue github.com/openai/math (family 025) claims c₁·log log b ≤ N(b) ≤ c₂·log log b for all large b, with unspecified constants. The catalogue lists the main theorem as formalised in Lean; the preprint has not yet been peer-reviewed.
- What can I try out in this app?
- You choose a numerator and denominator or an example and compare the greedy decomposition with the shortest one, as a pie and with a magnifier. For denominators up to 200 you see the hardest numerator and N(b). A panel shows shortest decompositions of 4/n for the Erdős–Straus conjecture.
Subject: Open problems, solved by AI
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