Egyptian Calculation (Decomposition into Unit Fractions with Three Terms)
Author: shiroeExecutions: 319
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Enter the fraction you want to check to see if it can be factored into the sum of three unit fractions.
Enter natural numbers for both the numerator and denominator. (Upper limits apply.)
The result will be displayed if a solution is found within the loop limit.
If no solution is found within the loop limit, only the number of loops
will be output. (The fact that no solution is found does not necessarily mean it is algorithmically impossible.
It also includes cases where the range of numbers is too large for manual calculation.
Furthermore, the fact that it cannot be decomposed into three unit fractions does not mean it has been “verified” as such.)
Enter natural numbers for both the numerator and denominator. (Upper limits apply.)
The result will be displayed if a solution is found within the loop limit.
If no solution is found within the loop limit, only the number of loops
will be output. (The fact that no solution is found does not necessarily mean it is algorithmically impossible.
It also includes cases where the range of numbers is too large for manual calculation.
Furthermore, the fact that it cannot be decomposed into three unit fractions does not mean it has been “verified” as such.)
| Number of loops | (Values of the inverse simplification coefficients leading to the decomposition) |
| Denominator of a unit fraction (1) | The denominator of the first term of a unit fraction expanded into three terms |
| Denominators of Unit Fractions (2) | The denominator of the second term in a unit fraction expanded into three parts |
| Denominator of a unit fraction (3) | The denominator of the third term in a unit fraction expanded into three terms |
R6 (2024). 10/11 (fr)_Ver.001. Displays up to 50 digits. Digit count can be adjusted. Real number calculation mode.
Plans: Considering factorization into four terms.
Plans: Considering factorization into four terms.
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This involves converting a fraction into a sum of unit fractions.
A unit fraction is a fraction with a numerator of 1.
Both the numerator and denominator of the fraction are natural numbers.
Here, we will consider the case of decomposing a fraction into three unit fractions.
The algorithm used is a reverse simplification loop; this type of algorithm
increases the flexibility of decomposition by repeatedly multiplying the numerator and denominator of the fraction being examined
by the same natural number.Although this is the reverse of prime factorization—where we construct composite numbers
to perform the search—as a way to mimic manual calculation,
simplifying by multiplying the numerator and denominator by natural numbers 1, 2, 3, … is a good method.
The results may differ from those of a sequential subtraction algorithm,
since the decomposition of unit fractions does not always yield a unique result.
(Note that the search range generally has no upper limit, so it is expected that the numbers handled may become too large
or that a timeout may occur.)