Weakly prime number
In number theory, a prime number is called weakly prime if it becomes not prime 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 base b weakly primes for b = 1 to 16 are: (sequence A186995 in the 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
- ↑ Weisstein, Eric W. "Weakly Prime". MathWorld.
- ↑ Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. Retrieved 18 February 2011.
- ↑ Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society. 91 (3). arXiv:0802.3361
. doi:10.1017/S1446788712000043. - ↑ Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. Retrieved 18 February 2011.