**FATHER PRIMES**

**“father” prime**shall be defined as one for which

**the sum of the squares of its digits**is also a

**prime**. The sum is therefore the

**“child” prime**.

**Example:**23 is a “father” prime because 2

^{2}+ 3

^{2}= 4 + 9 = 13. That is, 23 is the “father, (or progenitor)” of a prime child, namely 13. But, 13 is not a prime who can be a father (i.e. have a child), because 1

^{2}+ 3

^{2}= 1 + 9 = 10, a composite number.

191 to 83 to 73.

1499 to 179 to 131 to 11 to 2.

**Working Notes: (July 2001)**

All two-digit fathers are given here.

11: 1^{2} + 1^{2} = 1 + 1 = 2 prime

23: 2^{2} + 3^{2} = 4 + 9 = 13 prime

41: 4^{2} + 1^{2} = 16 + 1 = 17 prime

61: 6^{2} + 1^{2} = 36 + 1 = 37 prime

83: 8^{2} + 3^{2} = 64 + 9 = 73 prime

% % %

list of father-child primes: 100 < Prime fathers < 1000.

prime child | no. of fathers | prime fathers |

11 | 3 | 113, 131, 311 |

17 | 2 | 223, 401 |

19 | 2 | 313, 331 |

37 | 1 | 601 |

41 | 1 | 443 |

43 | 1 | 353 |

53 | 2 | 461, 641 |

59 | 3 | 137, 173, 317 |

61 | 2 | 463, 643 |

67 | 3 | 337, 373. 733 |

73 | 1 | 661 |

83 | 2 | 191, 911 |

89 | 1 | 229 |

97 | 1 | 409 |

101 | 2 | 467, 647 |

107 | 1 | 773 |

109 | 2 | 683, 863 |

113 | 1 | 449 |

131 | 4 | 179, 197, 719, 971 |

137 | 1 | 883 |

139 | 4 | 379, 397, 739, 937 |

149 | 1 | 829 |

163 | 3 | 199, 919, 991 |

179 | 2 | 797, 977 |

211 | 1 | 997 |

Total | 47 |
. |

113: 1^{2} + 1^{2} + 3^{2} = 1 + 1 + 9 = 11 prime

131: same result.

311: emirp for 113; is the 11th term in the list of odd prime sums

**137:** 1^{2} + 3^{2} + 7^{2} = 1 + 9 + 49 = **59 prime & all odd digits**

173: same

317: same

371: 7 x 53 [*but 3 ^{3} + 7^{3} + 1^{3} = 27 + 343 + 1 = 371*]

713: 23 x 31

731: 17 x 43

337: 3^{2} + 3^{2} + 7^{2} = 9 + 9 + 49 = 67

373: same

733: same

179: 1^{2} + 7^{2} + 9^{2} = 1 + 49 + 81 = 131

197: same

719: same

791: 7 x 113

917: 7 x 131

971: same

[note: 449 -(ssd)-> 113, a permutation of 131.]

379: 3^{2} + 7^{2} + 9^{2} = 9 + 49 + 81 = 139

397: same

739: same

793: 13 x 61

937: same

973: 7 x 139 [7’s co-factor is the “ssd” sum of this group]

199: 1^{2} + 9^{2} + 9^{2} = 1 + 81 + 81 = 163

919: same

991: same; emirp for 199.

Decade trios:

461 yields 53 641 yields 53

463 yields 61 643 yields 61

467 yields 101 647 yields 101

Strings:

179 gvs 131 gvs 11 gvs 2

191 gvs 83 gvs 73

463 gvs 61 gvs 37

443 gvs 41 gvs 17

111611 [or 611111] gvs 41 gvs 17

22441 [or 24421, or 44221] gvs 41 gvs 17

449 to 131 to 11 to 2

449 div by 81 = 5 r 44

so five 9’s, and 6, 2, & 2 could be used.

22699999/no

26299999/no

62299999/yes, a prime

**So 62299999 to 449 to 131 to 11 to 2 is another ancestral family tree of five generations.**

1699 gvs 199 gvs 163

**35466227 gvs 179 gvs 131 gvs 11 gvs 2 another family tree of 5 generations.**

1499 gvs 179 gvs 131 gvs 11 gvs 2; which is the one given at the top.

1499 — > 1 + 16 + 81 + 81 = 179

179 — > 1 + 49 + 81 = 131

131 — > 1 + 9 + 1 = 11

11 — > 1 + 1 = 2

1499 prime | 4199 no | 9149 no | 9419 prime |

1949 prime | 4919 prime | 9194 no | 9491 prime |

1994 no | 4991 no | 9914 no | 9941 prime |