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

1,1

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

Links

Table of n, a(n) for n=1..33.

Blair Kelly, Fibonacci and Lucas factorizations

Eric W. Weisstein, Sum of squares function

Example

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

Prog

(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.

Crossrefs

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

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

Sequence in context: A007645 A144919 A215801 * A015916 A023203 A086135

Adjacent sequences: A215806 A215807 A215808 * A215810 A215811 A215812

Keyword

nonn

Author

V. Raman, Aug 23 2012

Extensions

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

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

Status

approved