# Prime-free sequence problems

ramsey2879
For each the following recursive sequences,
find a number K>0 such that $${S_n}^2$$ + $$\bold{K}$$ is always composite.

1) $$S_0 = 10$$, $$S_1 = 11$$, $$S_n = 6S_{\left(n-1\right)}$$ - $$S_{\left(n-2\right)}$$

2) $$S_0 = 14$$, $$S_1 = 17$$, $$S_n = 6S_{\left(n-1\right)}$$ - $$S_{\left(n-2\right)}$$

Hint. Each problem has a separate K. Considering one of the series only, if you can find a second order equation in S_j and S_(j+1) that gives a constant for all j, and this equation also gives a second constant for the other series too, I would say that you are very near to solving these problems.

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