For example, 144 is not unprimeable, because changing the last digit into a nine we obtain 149, a prime. The number 200 is instead unprimeable (the smallest one), since none of the numbers 201, 203, 207, 205, and 209 are prime and all the other numbers which can be obtained from 200 (say, 300, or 270, or 208) are even, so they are not prime.
It is easy to prove that unprimeable numbers are infinite, since, for example, all the numbers of the form are unprimeable.
A prime which cannot be turned into another prime by changing a single digit is called weakly prime.
Up to there are 2855169 unprimeable numbers.