In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number.[1]

## Examples

The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 1012, followed by 6 (1102), 7 (1112) and 9 (10012).[2] The sequence of pernicious numbers begins

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, ... (sequence A052294 in the OEIS).

## Properties

No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a Hamming weight of one, and one is not considered to be a prime.[2] On the other hand, every number of the form ${\displaystyle 2^{n}+1}$ with ${\displaystyle n>1}$, including every Fermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number.[2]

A Mersenne number ${\displaystyle 2^{n}-1}$ has a binary representation consisting of ${\displaystyle n}$ ones, and is pernicious when ${\displaystyle n}$ is prime. Every Mersenne prime is a Mersenne number for prime ${\displaystyle n}$, and is therefore pernicious. By the Euclid–Euler theorem, the even perfect numbers take the form ${\displaystyle 2^{n-1}(2^{n}-1)}$ for a Mersenne prime ${\displaystyle 2^{n}-1}$; the binary representation of such a number consists of a prime number ${\displaystyle n}$ of ones, followed by ${\displaystyle n-1}$ zeros. Therefore, every even perfect number is pernicious.[3][4]

## Related numbers

• Odious numbers are numbers with an odd number of 1s in their binary expansion ().
• Evil numbers are numbers with an even number of 1s in their binary expansion ().

## References

1. ^ Deza, Elena (2021), Mersenne Numbers And Fermat Numbers, World Scientific, p. 263, ISBN 978-9811230332
2. ^ a b c Sloane, N. J. A. (ed.), "Sequence A052294", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
3. ^ Colton, Simon; Dennis, Louise (2002), "The NumbersWithNames Program", Seventh International Sumposium on Artificial Intelligence and Mathematics
4. ^ Cai, Tianxin (2022), Perfect Numbers And Fibonacci Sequences, World Scientific, p. 50, ISBN 978-9811244094