Problems & Puzzles:
S(n+1):=S(n)^2 - 2
Dmitry Kamenetsky poses the following
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)
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, ...
Dimitry sent his best solution: For K=7, S(1)=68987843381.