Problems & Puzzles: Puzzles

Puzzle 509. S(n+1):=S(n)^2 - 2 Dmitry Kamenetsky  poses the following puzzle: Let S be a sequence such that the first term S(1) is an odd prime and S(n+1):=S(n)^2 - 2. Let k be the largest number such that the first k terms of S are all prime. For example, if S(1)=3 then K is 4 since 3, 7, 47 and 2207 are all prime, but the next term 4870847 is not prime. Question: Find S that has the largest K. For simplicity report S(1) only. Carlos Rivera made a preliminary & quick exploration just to see how difficult this puzzle can be, and found just two more solutions for K>4: For K=5, S(1)= 1644103 For K=6, S(1)= 1383068639 So please send the smaller S(1) for K= 7, 8, ...

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Dimitry sent his best solution: For K=7, S(1)=68987843381.

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