An exercise in my number theory class asks me find the integer solutions of $x^2+xy - y^2 = x + y$. The exercise was divided in parts:
1) I have shown that $R = \mathbb{Z} + \frac{1 + \sqrt{5}}{2}\mathbb{Z}$ is a Euclidean domain. I did this with respect to the following function: $$\varphi: R\setminus\{0\} \to \mathbb{Z}_{\geq 0}: a+b\phi \mapsto a^2 + ab - b^2$$ where $\phi = \frac{1 + \sqrt{5}}{2}$.
2) I have shown that every unit in $R$ larger then $1$ is a power of $\phi$.
actual question: 3) In this part it is asked to show that if $p$ is a prime integer, then $p$ is either prime in $\mathbb{Z} + \phi\mathbb{Z}$ or it can be written as $(a + b\phi)(a + b - b\phi)$ for $a,b \in \mathbb{Z}$. I think I was able to do this. However, it is also asked to show that these two elements are not associated unless $p = 5$. For $p = 5$, I have found that they are indeed associated, but I am having a hard time proving it is not for other primes. I know that they both have the same norm, but I am not sure how to proceed.
EDIt: as @Hurkyl suggested, I could consider the ratio of both: $$\frac{a + b - b\phi}{a+b\phi} = \frac{(a+b - b\phi)^2}{a^2 + ab - b^2}$$ (by multiplying both numerator and denominator by $a+b - b\phi$). This gives me $$\frac{(a+b)^2 - 2(a+b)b\phi +b^2\phi^2}{a^2+ab - b^2}$$ and using that $\phi^2 = \phi + 1$, I find $$\frac{a^2 + 2ab + b^2 - 2ab\phi - 2b^2\phi +b^2\phi +b^2}{a^2+ab - b^2} = \frac{(a^2 + 2ab+2b^2) - (2ab + b^2)\phi}{a^2+ab - b^2}.$$ Since we consider $p = a^2 + ab - b^2$, we can rewrite this as $$\frac{(a^2 + 2ab + 2b^2) - (2ab + b^2)\phi}{p}$$ and here I am stuck...