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A215939
Prime numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.
1
2, 5, 11, 29, 41, 89, 131, 179, 331, 359, 401, 421, 431, 449, 509, 569, 571, 601, 631, 659, 691, 911
OFFSET
1,1
COMMENTS
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.
PROG
(PARI) forprime(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", ")))
CROSSREFS
Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).
Sequence in context: A288390 A127331 A040998 * A309857 A117719 A355516
KEYWORD
nonn,more
AUTHOR
V. Raman, Aug 27 2012
EXTENSIONS
Terms corrected by V. Raman, Sep 20 2012
STATUS
approved