Next Prime

The next prime function  gives the smallest prime larger than . The function can be given explicitly as

where  is the th prime and  is the prime counting function. For , 2, ... the values are 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, ... (OEIS A007918).

The next prime function is implemented in the Wolfram Language as NextPrime[n].

Finding the next prime after  gives the smallest -digit prime. For , 2, ..., the first few of these are 2, 11, 101, 1009, 10007, 100003, 1000003, ... (OEIS A003617). The amounts by which these exceed  are 1, 1, 1, 9, 7, 3, 3, 19, 7, 7, ... (OEIS A033873).

REFERENCES:

O'Hara, F. "The Smallest and Largest -Digit Primes for ." J. Recr. Math. 22, No. 4, 277-279, 1990.

Sloane, N. J. A. Sequences A003617/M2008, A007918, and A033873 in "The On-Line Encyclopedia of Integer Sequences."