When I found the extremely interesting and efficient (and free) program PARI-GP, which supports very efficiently unlimitedly large numbers and which can, among many other things, factorize and test primality really fast, certain prime numbers with a specific form started to interest me.

One interesting group of prime numbers is the one of the form
10^{n}+a, where n and a are integers and n>0 and a>0
(ie. for example numbers of the form 10^{n}+3 are 13, 103, 1003,
10003 and so on).

The table below shows all the values of n between 1 and 1000 and
a between 3 and 51 for which 10^{n}+a
is a prime number (ie. if n is 5 and a is 3, the number in question
is 100003).

a | n |
---|---|

3 | 1, 2, 5, 6, 11, 17, 18, 39, 56, 101, 105, 107, 123, 413, 426 |

7 | 1, 2, 4, 8, 9, 24, 60, 110, 134, 222, 412, 700, 999 |

9 | 1, 2, 3, 4, 9, 18, 22, 45, 49, 56, 69, 146, 202, 272 |

13 | 1, 2, 3, 17, 25, 81, 140, 142, 152, 280, 291, 406 |

19 | 1, 3, 5, 7, 10, 11, 17, 59, 81, 108, 574, 629 |

21 | 1, 3, 9, 17, 55, 77, 133, 195, 357 |

27 | 1, 2, 83, 167, 242 |

31 | 1, 2, 3, 14, 18, 44, 54, 89, 469 |

33 | 1, 3, 6, 9, 10, 31, 47, 70, 281, 366, 519, 532, 775 |

37 | 1, 2, 4, 6, 8, 13, 15, 39, 169, 184, 228, 255, 279, 632 |

39 | 2, 3, 4, 6, 8, 12, 20, 72, 196, 676 |

43 | 1, 5, 37, 253 |

49 | 1, 2, 3, 5, 8, 17, 24, 32, 65, 66, 67, 79, 83, 98 152, 260, 781 |

51 | 1, 2, 3, 13, 19, 81, 658 |

We can extend the idea above and search for primes in the form
b^{n}+a, where also b is an integer, and b>1 (the case b=1 would
be too trivial to be interesting, as it would just go through all the
primes regardless of the value of n). In the table below there are some
interesting values of these three parameters for which b^{n}+a is
prime:

b | a | n |
---|---|---|

2 | 1 | 1, 2, 4, 8, 16 |

3 | 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784 | |

5 | 1, 3, 5, 11, 47, 53, 141, 143, 191, 273, 341 | |

7 | 2, 4, 6, 8, 10, 16, 18, 20, 28, 30, 38, 44, 78, 88, 98, 126, 160, 174, 204, 214, 588, 610, 798, 926 | |

9 | 1, 2, 3, 5, 6, 7, 9, 10, 18, 23, 30, 37, 47, 57, 66, 82, 95, 119, 175, 263, 295, 317, 319, 327, 670, 697, 886 | |

3 | 2 | 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494 |

4 | 1, 2, 3, 6, 9, 10, 22, 30, 42, 57, 87, 174, 195, 198, 562, 994 | |

8 | 1, 2, 4, 5, 8, 13, 14, 20, 38, 44, 77, 88, 124, 152, 244, 557 | |

10 | 1, 2, 3, 6, 8, 18, 36, 98, 114, 134, 138, 212, 252, 516 | |

4 | 1 | 1, 2, 4, 8 |

3 | 1, 2, 3, 6, 8, 9, 14, 15, 42, 114, 195, 392, 555, 852 | |

7 | 1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 19, 22, 39, 44, 49, 63, 80, 87, 102, 107, 294, 305, 399, 463, 595, 599, 903, 944 | |

9 | 1, 3, 5, 9, 15, 33, 41, 335, 443, 671 | |

5 | 2 | 1, 3, 17, 143, 261, 551 |

4 | 2, 6, 10, 102, 494, 794 | |

6 | 1, 2, 3, 4, 13, 88, 177, 184, 297, 304, 310, 562, 892 | |

8 | 1, 95, 335 |