# Weakly prime number

In number theory, a prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit. Decimal digits are usually assumed.

The first weakly prime numbers are:

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, ... (sequence A050249 in the OEIS)

For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b number with n digits must produce (b−1) × n composite numbers when a digit is changed.

In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17×101000−17)/99 + 21686652. This is the largest known weakly prime number (As of 2011).

There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.

The smallest weakly prime base-b number for b = 2 to 10 is:

11111112 = 127
23 = 2
113114 = 373
3135 = 83
3341556 = 28151
4367 = 223
141038 = 6211
37389 = 2789
29400110 = 294001