Pernicious numbers are positive integers,
whose binary representation contains prime no. of 1s and the sum of binary digits
is a prime. First pernicious no. is 3 ->11_{2} -->1+1=2 which
is a prime. Next is 5 ->101_{2} -->1+1=2 which is a prime. Next is
6->110_{2} -->1+1=2 which is a prime. Then are 7,9,`10,
11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35,
36, 37, 38, 40, 41, 42, 44, 47, 48, 49, 50, 52, 55, 56, 59, 61, 62, 65,
66, 67, 68, 69, 70, 72, 73, 74, 76, 79, 80, 81, 82, 84, 87, 88, 91, 93,
94, 96, 97, 98, 100 etc.`
* No power of 2 is a
pernicious number since it is 1 followed by zeroes.
* Every even
Perfect Number is a pernicious number since they are of the form 2^{p-1}(2^{p}
-1 )where p is a prime no. Such no. are represented in binary as p no. of
1s and (p-1)no. of zeroes.
* A number of the
form (2^{p}
-1 )where p is a prime is a pernicious no. known as
**Pernicious Mersenne Number **whose binary representation
contains no zeroes. |