Problems & Puzzles: Puzzles

 Puzzle 1034. P+6k=Q, etc JM Bergot sent the following nice puzzle:   11+6=17 17+12=29 29+18=47 47+24=71 71+30=101 101+36=137 137+42=179 179+48=227 227+54=281 2810+60=341 FAIL This is a chain of nine successes. Starting with any prime and any multiple of 6, can you find a longer chain with increasing multiples of 6?  An example, 11+18=29, 29+24=53, 53+30=83, 83+36=119 FAIL.   Carlos Rivera, in a fast search found a chain of 21 terms: 23+6=29,...,1286+126=1409. Next term fails.   Later Bergot discovered that my improved solution was already published here.   Q. Find a larger chain starting with any prime and any multiple of 6.

During the week 10-16 Aproil 2021, contributions came from: Oscar Volpatti:

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Oscar wrote:

I found a chain of 22 successes by choosing:
p = 4745021,
6k = 3390.

It can't be extended further:
4745021 -  3384 = 4741637 = 313 x 15149
4745021 + 3390 = 4748411, prime

...
4817471 + 3516 = 4820987, prime

4820987 + 3522 = 4824509 = 277 x 17417

Full list of 23 primes:
4745021,  4748411,  4751807,  4755209,
4758617,  4762031,  4765451,  4768877,
4772309,  4775747,  4779191,  4782641,

4786097,  4789559,  4793027,  4796501,
4799981,  4803467,  4806959,  4810457,
4813961,  4817471,  4820987.

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