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Let $L_{n}$ be the $n$ th Lucas number. For example, $L_{1} = 1, L_{2} = 3, L_{3} = 4$.

Conjecture: there is no Gaussian primes in the sequence $(L_{n-1} + L_{n} i)$ for $n = 2$ to $\infty$.

I hope that this will be proved or disproved.

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$L_{n-1}+L_n i= (2+i)(F_n+F_{n-1}i)$.

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Your conjecture is true; the sum of the squares of the components is always divisible by 5. Indeed it is easy to show by induction that $L_{n+1}^2+L_n^2=5F_{2n}$ where $F_{2n}$ is the $2n^{th}$ Fibonacci number.

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