Weakly prime number

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Not to be confused with weak prime.

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 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 base b weakly primes for b = 1 to 16 are: (sequence A186995 in OEIS) [4]

111 = 2
11111112 = 127
23 = 2
113114 = 373
3135 = 83
3341556 = 28151
4367 = 223
141038 = 6211
37389 = 2789
29400110 = 294001
257311 = 3347
6B8AB7712 = 20837899
221613 = 4751
C371CD14 = 6588721
9880C15 = 484439
D2A4516 = 862789

References

  1. Weisstein, Eric W., "Weakly Prime", MathWorld.
  2. Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. Retrieved 18 February 2011. 
  3. Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society. 91 (3). arXiv:0802.3361Freely accessible. doi:10.1017/S1446788712000043. 
  4. Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. Retrieved 18 February 2011.