For example, 13 is not a weakly prime since changing its last digit into 7 we obtain another prime, 17. The smallest weakly prime is 294001, indeed changing any of its digits results in a composite number.
A composite number which cannot be turned into a prime by changing a single digit is called unprimeable.
Terence Tao has proved that there are infinite weakly primes.
A few extremely weakly primes are known, i.e., prime that become composite if any digit is removed, changed, or inserted. They are 40144044691, 58058453543, 89797181359, 185113489357, 213022025663, and 222498988079.