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Father Primes |
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FATHER PRIMES
A "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 22 + 32 = 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 12 + 32 = 1 + 9 = 10, a composite number.
An example of an ancestral line of fathers and children might be this:
191 to 83 to 73.
191 is father to 83 and grandfather to 73; 83 is father to 73; but 73 is "childless", as the sum (58) of the squares of its two digits (7 and 3) is a composite number. Hence, it is "the end of the line".
The longest ancestral lines so far established have 5 generations. One example of these is
1499 to 179 to 131 to 11 to 2.
Problem: Find a "father" and "grandfather" for 1499.
Working Notes: (July 2001)
All two-digit fathers are given here.
11: 12 + 12 = 1 + 1 = 2 prime
23: 22 + 32 = 4 + 9 = 13 prime
41: 42 + 12 = 16 + 1 = 17 prime
61: 62 + 12 = 36 + 1 = 37 prime
83: 82 + 32 = 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: 12 + 12 + 32 = 1 + 1 + 9 = 11 prime
131: same result.
311: emirp for 113; is the 11th term in the list of odd prime sums
137: 12 + 32 + 72 = 1 + 9 + 49 = 59 prime & all odd digits
173: same
317: same
371: 7 x 53 [but 33 + 73 + 13 = 27 + 343 + 1 = 371]
713: 23 x 31
731: 17 x 43
337: 32 + 32 + 72 = 9 + 9 + 49 = 67
373: same
733: same
179: 12 + 72 + 92 = 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: 32 + 72 + 92 = 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: 12 + 92 + 92 = 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 |