• Let F0=0,F1=1F_0 = 0, F_1 = 1 and Fn+1=Fn+Fn1 F_{{n+1}} = F_{n} + F_{n-1}, for all positive integers nn, be
    the Fibonacci sequence. Prove that for any positive integer mm there exist
    infi nitely many positive integers nn such that :
    Fn+2Fn+1+1Fn+2[m].F_n+2\equiv F_{n+1}+1 \equiv F_{n+2} \,\, [m].

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