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
- ↑ Weisstein, Eric W.. "Weakly Prime". http://mathworld.wolfram.com/WeaklyPrime.html.
- ↑ Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. http://www.primepuzzles.net/puzzles/puzz_017.htm.
- ↑ Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society 91 (3). doi:10.1017/S1446788712000043.
- ↑ Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. http://people.missouristate.edu/lesreid/Soln12.html.
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