= Achilles number
{c}
= Achilles
{c}
{synonym}
An <Achilles number> is a <powerful number> greater than one that is not a <perfect power>. In its <prime factorization>
$$
n=\prod_{i=1}^k p_i^{e_i},
$$
every <prime exponent> satisfies $e_i\geq2$, and $\gcd(e_1,\ldots,e_k)=1$. The first condition is precisely the definition of a <powerful number>. For the second, $n=m^r$ with <integers> $m,r>1$ exactly when $r$ divides every <prime exponent>; hence a <greatest common divisor> greater than one is precisely the obstruction.
For example, $72=2^3 3^2$ is an <Achilles number> because $\gcd(3,2)=1$. By contrast, $144=2^4 3^2=12^2$ is <powerful> but is a <perfect square>, so it is not an <Achilles number>.
Back to article page
