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%I #31 Oct 14 2019 03:33:21
%S 1,2,5,8,11,12,25,29,32,41,48,55,89,121,125,128,131,145,179,192,205,
%T 275,331,359,401,421,431,445,449,509,512,569,571,601,605,625,631,655,
%U 659,691,725,768,895,911,1025,1375
%N Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.
%C A number n can be written in the form a^2+5*b^2 if and only if n is 0, or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m) or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1), for integers i,j,k,m, for primes p,q.
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%o (PARI) for(i=2, 500, a=factorint(fibonacci(i))~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i", ")))
%Y Cf. A000045, A215939, A124132.
%Y Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).
%K nonn,more
%O 1,2
%A _V. Raman_, Aug 27 2012
%E Terms corrected by _V. Raman_, Sep 20 2012
%E a(46) from _Amiram Eldar_, Oct 14 2019
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