Problems & Puzzles: Puzzles

 Puzzle 384. Prime Reverse Keith numbers For sure you already know what a Keith number (KN) is. Then you know what is a Reverse Keith number (RKN), too. Here (A097060) you can find the RKN already known, as calculated by Jason Earls in Set. 2004: 12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611 From these 23 RKN, only 71 & 1593583 are primes. Questions 1. Can you extend the sequence A097060 ? 2. Can you extend it as far as to find two more primes in it?

Contributions came from B. Boncompagni, Antoine Verroken, Jacques Tramu & Anton Vrba who got the largest extension to the existing sequence.

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Anton wrote:

Yes I can extend the sequence of the 23 known RKN by a further 35 eight to eighteen digit RKNs which include only one eight digit prime!  Note there are no twelve digit RKNs .

26435142, 54734431, 57133931, 79112422, 89681171, 351247542, 428899438, 489044741, 578989902, 3207761244, 4156222103, 5426705064, 5785766973, 6336657062, 48980740972, 51149725354, 83626284302, 94183600081, 98665175305, 1935391095868, 6002181268035, 6334708806271, 12348924235856, 27488180694681, 76365591939888, 309217509306732, 352062080376812, 714692062325732, 723735537269331, 2437358882180001, 6792079280704301, 62244424802562056, 203414193894268461, 217049132946408803, 415499563488189604, 561624665953167171

Searching the 18 digit number  range took 3 hours on a Xeon 3.6GHz, program memory requirement being 1.8GByte for three sorted tables. The program was written in C++ using Visual Studio Express development environment.  Nineteen digits I am not attempting as signed 64 bit integer mathematics does not quite cover the required range.

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