Achilles was a powerful Trojan War hero who was extremely powerful but had one flaw–his achilles heel. Like him, Achilles numbers are powerful but not perfect. So, lets begin with a powerful number. A number is considered powerful if all of its prime factors remain factors once they are squared. 25 is a powerful number because its one prime factor, 5, remains a factor once its been squared (25, which goes into 25 once) Now lets move onto perfect powers, number that can be expressed as an integer power of another integer; 8 is a perfect power, as its 2 cubed. So now, back to the original premise – Achilles numbers are powerful, but they are not perfect powers. 72 is the first Achilles number, as it is powerful, but it is not a perfect prime. Others include 108, 200, 288, 392, 432, 500, and 648.