Problems & Puzzles: Puzzles

 Puzzle 400. 39883 Perhaps you already know the following curio: The smallest prime which can be represented as sum of a prime and its reversal in five different ways. [Gupta] As a matter of fact Gupta published the earliest prime of that type in k different ways for k = 1, 2, 3, 4 & 5: 383,  787, 1231, 36263 & 39883 The set of 5 primes associated to 39883by the definition are: 20981  22961  25931  26921  28901, that is to say: 39883=20981+18902=22961+19622=25931+13951=26921+12962=28901+10982 Due that I have not found published the solutions for k>5 I have computed them (again?). Here are them, for k=6, 7, 8, 9, 10 11, 12 13 & 14: 77267, 77267, 77267, 79687, 80897, 97879, 97879, 118691 & 119701. The set of primes 14 associated to 119701 are: 20399  23369  24359  25349  26339  27329  28319  41387  44357  46337  49307  82373  87323 & 89303 Questions. Can you extend this sequence for k>14?

Contributions came from J. K. Andersen, Anton Vrba, Farideh Firoozbakht, J. C. Rosa. All extended my initial list by far: Andersen (k=126), Anton (k=90), Farideh (k=31) & Rosa (k=61).

Here are the Andersen's results:

383, 787, 1231, 36263, 39883, 77267 (3), 79687, 80897, 97879 (2), 118691,
119701, 3380833 (2), 3507043 (2), 3590953 (2), 3680863, 3819073 (4),
3894883 (6), 7592957 (3), 7690967 (8), 7772777, 7782877, 7806977 (2),
7811077 (2), 7909087, 7984897 (2), 7990897 (10), 9782879 (9), 9896989 (15),
10111109 (5), 348929843 (4), 348989843 (17), 358948853 (6), 369170863 (4),
369767963, 378727873 (4).

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