Minimal prime (number theory)
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In number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime.
In base 10 there are exactly 26 minimal primes:
- 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (sequence A071062 in OEIS)
For example, 409 is a minimal prime because there is no prime among the shorter subsequences of the digits: 4, 0, 9, 40, 49, 09. The subsequence does not have to consist of consecutive digits, so 109 is not a minimal prime (because 19 is prime). But it does have to be in the same order
Similarly, there are exactly 32 composite numbers which have no shorter composite subsequence:
- 4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731 (sequence A071070 in OEIS)
[edit] References
- Chris Caldwell, The Prime Glossary: minimal prime, from the Prime Pages
- J. Shallit, Minimal primes, Journal of Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000
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