Problems & Puzzles: Puzzles

 Puzzle 903. Follow-up to Puzzle 902 Dmitry Kamenetsky sent the following follow-up to Puzzle 902: P={2, 3, 5}, Q={17, 13, 11} could be combined by the sumproduct operation giving a power b^n, n>1 of b>2. Example, b=5, n=7: (2 3) + (15619 15629) = 78125 = 5^7 Q. Find more pair of vectors P & Q of the same length k such that sumproduct(P,Q)=b^n for some b>2 & n>1.

Contribution came from Emmanuel Vantieghem. But first let's show the other results sent by Dmitry when he posed this puzzle.

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Dmitry sent:

(23 29) + (4783 4787) = 12^5
(163 167) + (751 757) = 12^5
(43 47 53) + (349 353 359) = 15^4
(83 89 97) + (307 311 313) = 17^4
(2 3 5 7) + (173 179 181 191) = 5^5
(23 29 31 37) + (2311 2333 2339 2341) = 6^7
(61 67 71 73) + (36749 36761 36767 36779) = 10^7
(19 23 29 31) + (367 373 379 383) = 14^4
(77849 77863 77867 77893 77899) + (78341 78347 78367 78401 78427) = 5^15
(13 17 19 23 29 31 37) + (4817 4831 4861 4871 4877 4889 4903) = 7^7
(5 7 11 13 17 19 23) + (19 23 29 31 37 41 43) = 15^3
(5 7 11 13 17 19 23 29 31) + (12553 12569 12577 12583 12589 12601 12611 12613 12619) = 5^9
(23 29 31 37 41 43 47 53 59) + (59 61 67 71 73 79 83 89 97) = 13^4
(1061 1063 1069 1087 1091 1093 1097 1103 1109) + (2437 2441 2447 2459 2467 2473 2477 2503 2521) = 17^6
(43 47 53 59 61 67 71 73 79 83 89 97 101 103 107) + (1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291) = 17^5

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Emmanuel wrote:

As in puzzle 902, the number of solutions is overwhelming.
Therefore I restricted myself to solutions in which the natural order of the primes is kept.

Some examples :
(2, 3) x (71, 73) = 19^2
(2, 3) x (191, 193) = 31^2
(2, 3) x (1373, 1381) = 83^2
(2, 3) x (3221, 3229) = 127^2
(2, 3) x (5573, 5581) = 167^2
(2, 3) x (6551, 6553) = 181^2
(2, 3) x (9767, 9769) = 221^2
(2, 3) x (9941, 9949) = 223^2
(2, 3, 5) x (431, 433, 439) = 66^2
(2, 3, 5) x (631, 641, 643) = 80^2
(2, 3, 5) x (1733, 1741, 1747) = 132^2
(2, 3, 5) x (9721, 9733, 9739) = 46^3
(2, 3, 5) x (10487, 10499, 10501) = 18^4
(2, 3, 5) x (35509, 35521, 35527) = 596^2
(2, 3, 5) x (50969, 50971, 50989) = 714^2
(2, 3, 5) x (52697, 52709, 52711) = 726^2
(2, 3, 5) x (65269, 65287, 65293) = 808^2
(2, 3, 5) x (90997, 91009, 91019) = 954^2
(3, 5, 7) x (257, 263, 269) = 63^2
(3, 5, 7) x (449, 457, 461) = 19^4
...

Even with this restriction, there are many solutions.
So I decided to retain the cases with exponent  n  > 2.
Examples :
(2, 3, 5) x (9721, 9733, 9739) = 46^3
(2, 3, 5) x (10487, 10499, 10501) = 18^4
(3, 5, 7) x (449, 457, 461) = 19^3
(13, 17, 19) x (85147, 85159, 85193) = 161^3
(19, 23, 29) x (1453, 1459, 1471) = 47^3
(19, 23, 29) x (3931, 3943, 3947) = 23^4
(43, 47, 53) x (349, 353, 359) = 15^4
(83, 89, 97) x (307, 311, 313) = 17^4
(83, 89, 97) x (331, 337, 347) = 45^3
(2, 3, 5, 7) x (173, 179, 181, 191) = 5^5
(2, 3, 5, 7) x (14683, 14699, 14713, 14717) = 63^3
(3, 5, 7, 11) x (1493, 1499, 1511, 1523) = 34^3
(3, 5, 7, 11) x (18233, 18251, 18253, 18257) = 78^3
(7, 11, 13, 17) x (27823, 27827, 27847, 27851) = 34^4
(19, 23, 29, 31) x (367, 373, 379, 383) = 14^4
(23, 29, 31, 37) x (107, 109, 113, 127) = 24^3
(23, 29, 31, 37) x (2311, 2333, 2339, 2341) = 6^7
(23, 29, 31, 37) x (79393, 79397, 79399, 79411) = 212^3
(29, 31, 37, 41) x (71, 73, 79, 83) = 22^3
(61, 67, 71, 73) x (36749, 36761, 36767, 36779) = 10^7
(97, 101, 103, 107) x (24097, 24103, 24107, 24109) = 56^4
(2, 3, 5, 7, 11) x (19661, 19681, 19687, 19697, 19699) = 82^3
(3, 5, 7, 11, 13) x (2017, 2027, 2029, 2039, 2053) = 43^3
(5, 7, 11, 13, 17) x (3467, 3469, 3491, 3499, 3511) = 57^3
(5, 7, 11, 13, 17) x (78707, 78713, 78721, 78737, 78779) = 161^3
(17, 19, 23, 29, 31) x (3533, 3539, 3541, 3547, 3557) = 75^3
(61, 67, 71, 73, 79) x (13883, 13901, 13903, 13907, 13913) = 47^4
(73, 79, 83, 89, 97) x (83023, 83047, 83059, 83063, 83071) = 327^3
(2, 3, 5, 7, 11, 13) x (77431, 77447, 77471, 77477, 77479, 77489) = 147^3
(3, 5, 7, 11, 13, 17) x (373, 379, 383, 389, 397, 401) = 28^3
(7, 11, 13, 17, 19, 23) x (17321, 17327, 17333, 17341, 17351, 17359) = 116^3
(23, 29, 31, 37, 41, 43) x (10181, 10193, 10211, 10223, 10243, 10247) = 38^4
(73, 79, 83, 89, 97, 101) x (29201, 29207, 29209, 29221, 29231, 29243) = 248^3
(79, 83, 89, 97, 101, 103) x (103, 107, 109, 113, 127, 131) = 40^3
(2, 3, 5, 7, 11, 13, 17) x (62927, 62929, 62939, 62969, 62971, 62981, 62983) = 154^3
(5, 7, 11, 13, 17, 19, 23) x (19, 23, 29, 31, 37, 41, 43) = 15^3
(13, 17, 19, 23, 29, 31, 37) x (4817, 4831, 4861, 4871, 4877, 4889, 4903) = 7^7
(37, 41, 43, 47, 53, 59, 61) x (131, 137, 139, 149, 151, 157, 163) = 37^3
(73, 79, 83, 89, 97, 101, 103) x (3571, 3581, 3583, 3593, 3607, 3613, 3617) = 131^3
(89, 97, 101, 103, 107, 109, 113) x (92377, 92381, 92383, 92387, 92399, 92401, 92413) = 405^3
(5, 7, 11, 13, 17, 19, 23, 29) x (419, 421, 431, 433, 439, 443, 449, 457) = 38^3
(7, 11, 13, 17, 19, 23, 29, 31) x (53047, 53051, 53069, 53077, 53087, 53089, 53093, 53101) = 24^5
(11, 13, 17, 19, 23, 29, 31, 37) x (82633, 82651, 82657, 82699, 82721, 82723, 82727, 82729) = 246^3
(5, 7, 11, 13, 17, 19, 23, 29, 31) x (12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619) = 5^9
(11, 13, 17, 19, 23, 29, 31, 37, 41) x (15527, 15541, 15551, 15559, 15569, 15581, 15583, 15601, 15607) = 151^3
(23, 29, 31, 37, 41, 43, 47, 53, 59) x (59, 61, 67, 71, 73, 79, 83, 89, 97) = 13^4
(37, 41, 43, 47, 53, 59, 61, 67, 71) x (113, 127, 131, 137, 139, 149, 151, 157, 163) = 41^3
(37, 41, 43, 47, 53, 59, 61, 67, 71) x (82729, 82757, 82759, 82763, 82781, 82787, 82793, 82799, 82811) = 341^3
(41, 43, 47, 53, 59, 61, 67, 71, 73) x (89797, 89809, 89819, 89821, 89833, 89839, 89849, 89867, 89891) = 359^3
(7, 11, 13, 17, 19, 23, 29, 31, 37, 41) x (1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801) = 74^3
(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) x (199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263) = 34^3
(13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53) x (43651, 43661, 43669, 43691, 43711, 43717, 43721, 43753, 43759, 43777, 43781) = 249^3
(31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73) x (91823, 91837, 91841, 91867, 91873, 91909, 91921, 91939, 91943, 91951, 91957) = 377^3

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