# Why are they called 'pernicious' numbers?

The definition of a pernicious number:

In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.

The meaning of 'pernicious':

Intriguing. How did pernicious numbers end up being called that?

• Maybe this is in the sprit of "odious" and "evil" numbers (those with an odd, resp. even number of ones in their binary representation)? – Michael Stoll Jan 24 '15 at 11:41
• Let $t(n)$ be the digit sum of the binary representation of $n$. Then Google will tell you that $n$ is called odious if $t(n)$ is odd and evil if $t(n)$ is even. Thus every number is either odious or evil, and therefore the words "odious" and "evil" cannot be pejorative in this context. It seems very likely that "pernicious" was chosen in the same spirit because it starts with the letter "p". – Steven Landsburg Jan 24 '15 at 11:42
• It looks like Michael Stoll posted while I was typing. – Steven Landsburg Jan 24 '15 at 11:42
• Since "odd" gives rise to "odious", and "even" to "evil", "prime" should really give rise to something with negative connotations beginning with "pr", not just with "p". "Primitive"? "Promiscuous"? "Privileged"? "Prevaricating"? – Gerry Myerson Mar 18 '18 at 22:57

Let $t\,(n)$ be the digit sum of the binary representation of $n$. Then Google will tell you that $n$ is called odious if $t\,(n)$ is odd and evil if $t\,(n)$ is even. Thus every number is either odious or evil, and therefore the words "odious" and "evil" cannot be pejorative in this context. It seems very likely that "pernicious" was chosen in the same spirit because it starts with the letter "p".