View source for Weakly prime number

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In [[Number theory|number theory]], a [[Prime number|prime number]] is called '''weakly prime''' if it becomes [[Composite number|composite]] when any one of its digits is changed to every single other digit.<ref>{{MathWorld|urlname=WeaklyPrime|title=Weakly Prime}}</ref> [[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, ... {{OEIS|A050249}}

For the first of these, each of the 54 numbers '''0'''94001, '''1'''94001, '''3'''94001, ..., 29400'''9''' are composite. A weakly prime [[Radix|base-''b'']] number with ''n'' digits must produce (''b''−1)&nbsp;×&nbsp;''n'' composite numbers when a digit is changed.

In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17{{e|1000}}−17)/99 + 21686652.<ref>{{cite web |url=http://www.primepuzzles.net/puzzles/puzz_017.htm |title=Puzzle 17 – Weakly Primes |author=Carlos Rivera |work=The Prime Puzzles & Problems Connection |access-date=18 February 2011}}</ref> This is the largest known weakly prime number {{as of|2011|lc=on}}.

There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.<ref>{{cite journal|author=[[Biography:Terence Tao|Terence Tao]]|title=A remark on primality testing and decimal expansions|journal=Journal of the Australian Mathematical Society|year=2011|volume=91|issue=3|arxiv=0802.3361|doi=10.1017/S1446788712000043}}</ref>

The smallest weakly prime base-''b'' number for ''b'' = 2 to 10 is:<ref>{{cite web |url=http://people.missouristate.edu/lesreid/Soln12.html |title=Solution to Problem #12 |author=Les Reid |work=Missouri State University's Problem Corner |access-date=18 February 2011}}</ref>
:1111111<sub>2</sub> = 127
:2<sub>3</sub> = 2
:11311<sub>4</sub> = 373
:313<sub>5</sub> = 83
:334155<sub>6</sub> = 28151
:436<sub>7</sub> = 223
:14103<sub>8</sub> = 6211
:3738<sub>9</sub> = 2789
:294001<sub>10</sub> = 294001

== References ==
{{Reflist}}

[[Category:Classes of prime numbers]]
[[Category:Base-dependent integer sequences]]

{{Sourceattribution|Weakly prime number|1}}