Let $\{F_n\} -$ Fibonacci sequence: $F_1=F_2=1, F_{n+1}=F_n+F_{n-1}, n\ge2$. Prove that $$x^2-x-1 = F_{12m + 7}$$ has no solutions. $x\in \mathbb N$
My work so far:
$$F_{12m+7}\equiv5(\bmod8),$$ then $x\equiv3(\bmod8)$ or $x\equiv6(\bmod8)$
2026-03-30 02:06:49.1774836409
Prove that $x^2-x-1 = F_{12m + 7}$ has no solutions.
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