For A,B∈Z\documentclass[12pt]{minimal}
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\begin{document}$$A,B\in {\mathbb {Z}}$$\end{document}, the Lucas sequence un(A,B)(n=0,1,2,…)\documentclass[12pt]{minimal}
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\begin{document}$$u_n(A,B)\ (n=0,1,2,\ldots )$$\end{document} are defined by u0(A,B)=0\documentclass[12pt]{minimal}
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\begin{document}$$u_0(A,B)=0$$\end{document}, u1(A,B)=1\documentclass[12pt]{minimal}
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\begin{document}$$u_1(A,B)=1$$\end{document}, and un+1(A,B)=Aun(A,B)-Bun-1(A,B)(n=1,2,3,…).\documentclass[12pt]{minimal}
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\begin{document}$$u_{n+1}(A,B)=Au_n(A,B)-Bu_{n-1}(A,B)\ (n=1,2,3,\ldots ).$$\end{document} For any odd prime p and positive integer n, we establish the new result upn(A,B)-A2-4Bpun(A,B)pn∈Zp,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{u_{pn}(A,B)-\left( \frac{A^2-4B}{p}\right) u_n(A,B)}{pn}\in {\mathbb {Z}}_p, \end{aligned}$$\end{document}where ·p\documentclass[12pt]{minimal}
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\begin{document}$$\left( \frac{\cdot }{p}\right) $$\end{document} is the Legendre symbol and Zp\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_p$$\end{document} is the ring of p-adic integers.
