Egyptian Calculation (Decomposition into Unit Fractions with 4 Terms)
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The results may differ from those obtained using a sequential subtraction algorithm.
This is because the results of unit fraction decomposition are not always unique.
(Note: The search range generally has no upper limit, so if the numbers become too large,
a timeout may occur.)
Enter the fraction you want to check to see if it can be decomposed into the sum of four unit fractions.
Enter natural numbers for both the numerator and denominator. (Upper limits apply.)
The result is output 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 four unit fractions does not mean it has been “verified” as impossible.)
This is because the results of unit fraction decomposition are not always unique.
(Note: The search range generally has no upper limit, so if the numbers become too large,
a timeout may occur.)
Enter the fraction you want to check to see if it can be decomposed into the sum of four unit fractions.
Enter natural numbers for both the numerator and denominator. (Upper limits apply.)
The result is output 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 four unit fractions does not mean it has been “verified” as impossible.)
| 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 four terms |
| Denominator of a unit fraction (2) | The denominator of the second term in a unit fraction expanded into four parts |
| Denominator of a unit fraction (3) | The denominator of the third term of a unit fraction expanded into four terms |
| Denominator of a unit fraction 4) | The denominator of the fourth term in a unit fraction expanded 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 four 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 involves creating composite numbers—the reverse of prime factorization—
to conduct the search, as a way to mimic manual calculations,
simplifying by multiplying both the numerator and denominator of the fraction by natural numbers 1, 2, 3, … is a good method.