Let (Fn)n≥1\documentclass[12pt]{minimal}
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\begin{document}$$(F_n)_{n \ge 1}$$\end{document} be the sequence of Fibonacci numbers. For all integers a and b≥1\documentclass[12pt]{minimal}
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\begin{document}$$b \ge 1$$\end{document} with gcd(a,b)=1\documentclass[12pt]{minimal}
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\begin{document}$$\gcd (a, b) = 1$$\end{document}, let [a-1modb]\documentclass[12pt]{minimal}
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\begin{document}$$[a^{-1} \!\bmod b]$$\end{document} be the multiplicative inverse of a modulo b, which we pick in the usual set of representatives {0,1,⋯,b-1}\documentclass[12pt]{minimal}
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\begin{document}$$\{0, 1, \dots , b-1\}$$\end{document}. Put also [a-1modb]:=∞\documentclass[12pt]{minimal}
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\begin{document}$$[a^{-1} \!\bmod b]:= \infty $$\end{document} when gcd(a,b)>1\documentclass[12pt]{minimal}
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\begin{document}$$\gcd (a, b) > 1$$\end{document}. We determine all positive integers m and n such that [Fm-1modFn]\documentclass[12pt]{minimal}
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\begin{document}$$[F_m^{-1} \bmod F_n]$$\end{document} is a Fibonacci number. This extends a previous result of Prempreesuk, Noppakaew, and Pongsriiam, who considered the special case m∈{3,n-3,n-2,n-1}\documentclass[12pt]{minimal}
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\begin{document}$$m \in \{3, n - 3, n - 2, n - 1\}$$\end{document} and n≥7\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 7$$\end{document}. Let (Ln)n≥1\documentclass[12pt]{minimal}
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\begin{document}$$(L_n)_{n \ge 1}$$\end{document} be the sequence of Lucas numbers. We also determine all positive integers m and n such that [Lm-1modLn]\documentclass[12pt]{minimal}
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\begin{document}$$[L_m^{-1} \bmod L_n]$$\end{document} is a Lucas number.
