From Wikipedia

"In number theory, a pernicious number is a positive integer that has a number of 1s in its binary representation which is a prime number. Equivalently, the sum of the digits of a pernicious number, when represented in base 2, is a prime.

This seemingly holds true for all base 2 numbers where the first bit is set to true (1), however, the same cannot be said for any other base 2 representation where the first bit is false (0). For example:

0111 = 07

1101 = 13

1011 = 11


1110 = 14

0010= 02

0100= 04

Is it fair to say that wikipedia is incorrect or am I misinterpreting the statement?

The statement to me implies that the sum of the the true (1) values with in a pernicious number will always be prime. As shown however this can only be true for numbers with a true (1) first bit. My thoughts are that either the definition of pernicious requires that the sum of the digits be prime as well as the count of true (1) bits being prime or that the statement is incorrect.

  • 2
    $\begingroup$ Can you separate your own thoughts from the Wikipedia quote? $\endgroup$ – Ilya Mar 22 '13 at 16:38
  • $\begingroup$ It seems that you write binary in reverse from the usual order. $\endgroup$ – Alfonso Fernandez Mar 22 '13 at 16:40
  • $\begingroup$ Sorry about that, reordered the digits. $\endgroup$ – CBusBus Mar 22 '13 at 16:42
  • $\begingroup$ I've updated the question as requested. $\endgroup$ – CBusBus Mar 22 '13 at 16:47
  • $\begingroup$ Since the binary representation contains only $0$'s and $1$'s, the number of $1$'s and the sum of the digits are the same; also, $1$ is not prime, and $10101 = 21$ is an example of a composite pernicious number ending with $1$. $\endgroup$ – Alfonso Fernandez Mar 22 '13 at 16:52

Wikipedia is just defining the pernicious numbers as a subset of the naturals. They are given in OEIS A052294. There is no requirement that the number be prime or composite, just that the number of $1$ bits be prime. In your table $14$ is pernicious but not prime, $2$ is prime but not pernicious, and $4$ is neither.


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