How to Find a K for Prime-Free Recursive Sequences?

[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.

 

[AKG]

Uh, K = 0?

S_n² + K
= S_n² + 0
= S_n²
= (S_n)(S_n)

I'm sure that was more steps than was required to convince you, but a square number is obviously always composite. Maybe you meant simply that S_n is always composite, or maybe S_(n²) as opposed to (S_n)²?

 

[ramsey2879]

One more condition! K>0! Sorry