A124130 Numbers n such that L_n = a^2 + b^2, where L_n is the n-th Lucas number with a and b integers.

0, 1, 3, 6, 7, 13, 19, 30, 31, 37, 43, 49, 61, 67, 73, 78, 79, 91, 111, 127, 150, 163, 169, 183, 199, 223, 307, 313, 349, 361, 390, 397, 433, 511, 523, 541, 606, 613, 619, 709, 750, 823, 907, 1087, 1123, 1129, 1147, 1213, 1279, 1434

Offset: 1

Author

Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006

Keywords

• nonn

Comments

Congruence considerations eliminate many indices, but the remaining numbers were factored. These have no prime factors of the form p=4m+3 dividing them to an odd power. Joint work with Kevin O'Bryant and Dennis Eichhorn.

To write Lucas(n) as a^2+b^2: find the a^2+b^2 representation for the individual prime factors by Cornacchia's algorithm, and merge them by using the formulas (a^2+b^2)(c^2+d^2) = |ac+bd|^2 + |ad-bc|^2 = |ac-bd|^2 + |ad+bc|^2. - V. Raman, Oct 04 2012

Values of A000032(n) such that A000032(n) or A000032(n)/2 is a square are only 1, 2, 4, 18. So a and b must be distinct and nonzero for all values of this sequence except 0, 1, 3, 6. - Altug Alkan, May 04 2016

1501 <= a(51) <= 1531. 1531, 1651, 1747, 1758, 1849, 1950, 1951, 2053, 2413, 2449, 2467, 3030, 4069, 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

			a(5) = 13 because the first five Lucas numbers that are the sum of two squares are L_1, L_3, L_6, L_7 and L_13 = 521 = 11^2 + 20^2.
		

Links

• Blair Kelly, Fibonacci and Lucas factorizations
• Eric Weisstein's World of Mathematics, Sum of squares function
• Wikipedia, Cornacchia's algorithm

Crossrefs

Intersection of A000032 or A000204 = Lucas numbers and A001481.

Cf. A215809, A215906, A215907.

Programs

Mathematica

Select[Range[0, 200], SquaresR[2, LucasL[#]] > 0&] (* T. D. Noe, Aug 24 2012 *)

PARI

for(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", "))) \\ V. Raman, Aug 23 2012

Python

from itertools import count, islice
from sympy import factorint, lucas
def A124130_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(lucas(n)).items()),count(0))
A124130_list = list(islice(A124130_gen(),20)) # Chai Wah Wu, Jun 27 2022

Extensions

a(1)=0 and a(26)-a(45) from V. Raman, Sep 06 2012

a(46)-a(50) from Chai Wah Wu, Jul 22 2020