A215938 Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 5*b^2.

1, 2, 5, 8, 11, 12, 25, 29, 32, 41, 48, 55, 89, 121, 125, 128, 131, 145, 179, 192, 205, 275, 331, 359, 401, 421, 431, 445, 449, 509, 512, 569, 571, 601, 605, 625, 631, 655, 659, 691, 725, 768, 895, 911, 1025, 1375

Offset: 1

Author

V. Raman, Aug 27 2012

Keywords

• nonn
• more

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.

Links

• Blair Kelly, Fibonacci and Lucas factorizations

Crossrefs

Cf. A000045, A215939, A124132.

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

Programs

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", ")))

Extensions

Terms corrected by V. Raman, Sep 20 2012

a(46) from Amiram Eldar, Oct 14 2019