Enter a number to check if it is an Achilles number. Or input a start and end range to generate all Achilles numbers within that range.
An Achilles number is a special type of strong non-perfect power. Specifically: An Achilles number is the product of multiple prime factors, where the exponent of each prime factor is at least 2. However, the number itself cannot be expressed as an integer power of another number (i.e., it is not a perfect square, cube, etc.).
For example: 72 is an Achilles number because \( 72 = 2^3 \times 3^2 \). All exponents are greater than 1. And 72 is not a perfect power (it cannot be written as the square or cube of an integer).
To check if a number is an Achilles number:
Solution:
1. Prime Factorization:
\( 288 = 2^5 \times 3^2 \).
2. Verify Exponents:
The exponents (5 and 2) are greater than 1.
3. Confirm Strong Non-Perfect Power Condition:
288 cannot be expressed as an integer power of another number.
Result: 288 is an Achilles number.
Solution:
1. Prime Factorization:
\( 360 = 2^3 \times 3^2 \times 5^1 \).
2. Verify Exponents:
The exponent of 5 is 1, which does not meet the criteria.
Result: 360 is not an Achilles number.
Solution:
1. Prime Factorization:
\( 784 = 2^4 \times 7^2 \).
2. Verify Exponents:
The exponents (4 and 2) are greater than 1.
3. Confirm Strong Non-Perfect Power Condition:
\( 784 = 28^2 \), making it a perfect square.
Result: 784 is not an Achilles number.