A215809 Prime numbers n for which the Lucas number L(n) (see A000032) is the sum of two squares.

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 127, 163, 199, 223, 307, 313, 349, 397, 433, 523, 541, 613, 619, 709, 823, 907, 1087, 1123, 1129, 1213, 1279, 1531

Offset: 1

Author

V. Raman, Aug 23 2012

Keywords

• nonn

Comments

These Lucas numbers L(n) have no prime factor congruent to 3 mod 4 to an odd power.

Also prime numbers n such that the Lucas number L(n) can be written in the form a^2 + 5*b^2.

Any prime factor of Lucas(n) for n prime is always of the form 1 (mod 10) or 9 (mod 10).

A number n can be written in the form a^2+5*b^2 (see A020669) 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.

1607 <= a(34) <= 1747. 1747, 1951, 2053, 2467, 5107, 5419, 5851, 7243, 7741, 8467, 13963, 14449, 14887, 15511, 15907, 35449, 51169, 193201, 344293, 387433, 574219, 901657, 1051849 are terms. - Chai Wah Wu, Jul 22 2020

Examples

			Lucas(19) = 9349 = 95^2 + 18^2.
Lucas(19) = 9349 = 23^2 + 5*42^2.
		

Links

• Blair Kelly, Fibonacci and Lucas factorizations
• Eric W. Weisstein, Sum of squares function

Crossrefs

Cf. A000032, A001481, A124130, A180363, A215906, A215907.

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

Programs

PARI

forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%4==3&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", "))) \\ a^2+b^2 form.

PARI

forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; 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", "))) \\ a^2+5*b^2 form.

Extensions

Merged A215941 into this sequence, T. D. Noe, Sep 21 2012

a(30)-a(33) from Chai Wah Wu, Jul 22 2020