Father Primes

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
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