A215938 -id:A215938 - cp's OEIS frontend

A215939 Prime numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

2, 5, 11, 29, 41, 89, 131, 179, 331, 359, 401, 421, 431, 449, 509, 569, 571, 601, 631, 659, 691, 911

Offset: 1

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V. Raman, Aug 27 2012

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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.

Blair Kelly, Fibonacci and Lucas factorizations

Cf. A000045, A215938, A124132.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

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", ")))

Terms corrected by V. Raman, Sep 20 2012

cp's OEIS Frontend

A215939 Prime numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

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