Minimal prime (number theory)

Minimal prime (number theory)

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 OEIS|id=A071062.

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; so, for example, 991 is still a minimal prime even though a subset of the digits can form the shorter prime 19 by changing the 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 OEIS|id=A071070.

References

*Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=MinimalPrime "The Prime Glossary: minimal prime"] , from the Prime Pages
*J. Shallit, [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.ps "Minimal primes"] , "Journal of Recreational Mathematics", vol. 30.2, pp. 113-117, 1999-2000


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