My problem is a little more difficult. However, essentially, my problem comes up when we're supposed to find the solution to:
$R(n)=R(n-3)+1$, $R(0)=R(1)=R(2)=1$ (the initial conditions don't matter too much in terms of my question)
Then we first solve the homogeneous system $R(n)=R(n-3)$, and we solve the characteristic polynomial $\lambda^n-\lambda^{n-3}=0$ which implies we're being asked to solve $\lambda^3-1=0$ (to find the non-zero solutions). Under any field, this factors as $\lambda^3-1 =(\lambda-1)(\lambda^2+\lambda+1)$. But under the field, say, $F_5=\mathbb{Z}/5\mathbb{Z}=\{0,1,2,3,4\}$, the $f(x)=x^2 +x+1$ is irreducible. Therefore, what does this say about the solutions $\lambda_1, \lambda_2, \lambda_3$ being the bases of the exponential terms in the solution: $R(n)=a(\lambda_1)^n + b(\lambda_2)^n + c(\lambda_3)^n + P(n)$ (where $P(n)$ is the polynomial that makes the solution satisfy the non-homogeneous recursion).
Thank you in advance :)