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COMMENTS
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Indices for Lucas-Fibonacci prime twins are A080327[n] = {4,5,7,11,13,17,47,...}. Corresponding Fibonacci-Lucas prime twins are A121533[n] = {3,5,13,89,233,1597,2971215073,...}. Probable primes Fibonacci[148091] and Lucas[148091] are the next probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits.
General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka, Sep 02 2008]
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MATHEMATICA
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Do[f=Fibonacci[n]; l=Fibonacci[n-1]+Fibonacci[n+1]; If[PrimeQ[f]&&PrimeQ[l], Print[{f, l}]], {n, 10000}]
nn=1000; Transpose[Select[Thread[{Fibonacci[Range[nn]], LucasL[ Range[nn]]}], And@@PrimeQ[#]&]][[2]] (* From Harvey P. Dale, Jul 08 2011 *)
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