Achilles number

Given a composite numberMathworldPlanetmath n with factorization


where the pi are all distinct primes, and the ai are positive integers (and also distinct, always ai>1), and ω(x) is the number of distinct prime factors function, then n is a squarefull number, and if it is also the case that there is no solution to n=xy in integers (with xn and y>1), then n is called an Achilles number.

So, for example, 1323, being 33×72, is divisible by the squares of both 3 and 7, but its square rootMathworldPlanetmath is approximately 36.373, its cubic root is about 10.9779, its fourth root is about 6.031, its fifth root is about 4.21, its sixth root is 3.313, seventh root about 2.792, etc.

An Achilles number has to have distinct exponents for its prime factorsMathworldPlanetmath: with equal exponents, we can see that paxpbx=(papb)x. The reverse is often not true: for example, 144 is 24×32, yet it can be expressed as 122.

According to Greek legend, Achilles was a powerful, seemingly invincible warrior who fought in the Trojan War. In some versions of the story, his mother Thetis held baby Achilles by the left heel and dipped him into the Styx river, making him invulnerable to attack anywhere on his body except the one heel. Thus, Achilles was very powerful, but not perfect because of his fatal weak spot. Likewise, Achilles numbers are powerfull but they are not perfect powersMathworldPlanetmath.


Title Achilles number
Canonical name AchillesNumber
Date of creation 2013-03-22 17:10:33
Last modified on 2013-03-22 17:10:33
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A51