# 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.[1] 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.[2] 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.[3]

The smallest weakly prime base-b number for b = 2 to 10 is:[4]

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

## References

1. Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection.
2. Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society 91 (3). doi:10.1017/S1446788712000043.
3. Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner.