#### LifeIn

##### Well-known member

11 (base 10) = 23 (base 4), and sure enough, 23 (as a base 10 number) is prime, and larger than 11.

13 (base 10) = 31 (base 4), and sure enough, 31 is prime as a base 10 number.

17 (base 10) = 101 (base 4), and 101 is in fact prime as a base 10 number.

19 (base 10) = 103 (base 4), which is prime as a base 10 number.

23 (base 10) = 113 (base 4), which is prime as a base 10 number.

29 (base 10) = 131 (base 4), which is prime as a base 10 number.

This continues to work for 31 (producing 133), and 37 (producing 211).

However it fails at 41 which produces 221 which is not prime. It is not surprising that the method does eventually fail. The only thing that might be a little surprising is why it worked as well as it did all the way up to 37. Primes are not very dense around 211, 133, 131,113, etc. The fact that all these numbers which have no reason to be prime are in fact prime is surprising. So, what is the reason that this method "almost" worked for so many numbers?