Egyptian Fraction Calculator

a/b = 1/d₁ + 1/d₂ + ...

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About Egyptian Fraction Calculator

An Egyptian fraction calculator decomposing a/b into distinct unit fractions 1/d₁ + 1/d₂ + ... using the greedy algorithm. Shows step-by-step decomposition, verifies the sum, and displays alternative representations. All client-side.

Egyptian Fraction Calculator Features

  • Greedy decomposition
  • Step-by-step
  • Verification
  • Alternative reps
  • History
Egyptian fraction: sum of distinct unit fractions. Ancient Egyptians wrote fractions this way (except 2/3). Greedy algorithm (Fibonacci-Sylvester): subtract largest 1/d ≤ a/b, repeat. Always terminates. Example: 4/17 = 1/5 + 1/29 + 1/1233 + 1/3039345.

How to Use

Enter fraction a/b:

  • a: Numerator
  • b: Denominator
  • Result: Sum of unit fractions

Greedy Algorithm

For a/b: find smallest d with 1/d ≤ a/b (d = ⌈b/a⌉). Subtract: a/b − 1/d = (ad−b)/(bd). Repeat with new fraction. The Erdős-Straus conjecture: 4/n = 1/x+1/y+1/z always has a solution.

History

Rhind Papyrus (1650 BCE) contains an Egyptian fraction table. Egyptians used unit fractions exclusively (plus 2/3). The greedy algorithm was described by Fibonacci (1202) and Sylvester (1880).

Step-by-Step Instructions

  1. 1Enter numerator.
  2. 2Enter denominator.
  3. 3View decomposition.
  4. 4Check verification.
  5. 5See steps.

Egyptian Fraction Calculator — Frequently Asked Questions

Does the greedy algorithm always terminate?+

Yes! Each step strictly reduces the numerator of the remainder. Since numerators are positive integers, the process must terminate. However, the denominators can grow very rapidly — for 4/17, the greedy algorithm produces denominators up to 3,039,345.

Is the greedy decomposition unique?+

No! Most fractions have multiple Egyptian fraction representations. 2/3 = 1/2+1/6 = 1/3+1/4+1/12. The greedy algorithm gives one specific representation, but there may be shorter or more elegant ones.

What is the Erdős-Straus conjecture?+

For every n≥2, 4/n can be written as 1/x+1/y+1/z with positive integers x,y,z. Verified for all n up to 10¹⁴. One of the most famous open problems in Egyptian fraction theory.

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