A236264 Even indices of Fibonacci numbers which are the sum of two squares.

0, 2, 6, 12, 14, 26, 38, 62, 74, 86, 98, 122, 134, 146, 158, 182, 222, 254, 326, 338, 366, 398, 446, 614, 626, 698, 722, 794, 866, 1022, 1046, 1082, 1226, 1238, 1418, 1646, 1814, 2174, 2246, 2258, 2294, 2426, 2558

Offset: 1

Author

Jean-François Alcover, Jan 21 2014

Keywords

• nonn
• more

Comments

The first 10 such Fibonacci numbers are 0, 1, 8, 144, 377, 121393, 39088169, 4052739537881, 1304969544928657, 420196140727489673.

Ballot & Luca (Proposition 1) show that this sequence has asymptotic density 0. - Charles R Greathouse IV, Jan 21 2014

a(43) >= 2558. Determining this term requires factoring the Lucas number L_1279. - Charles R Greathouse IV, Jan 21 2014

3002 <= a(44) <= 3302. 3302, 3698, 4898 are terms. - Chai Wah Wu, Jul 23 2020

Examples

			Fibonacci(14) = 377 = 19^2 + 4^2, so 14 is in the sequence.
		

Links

• Christian Ballot and Florian Luca, On the equation x^2+dy^2=Fn, Acta Arith. 127 (2007), 145-155.
• Kevin O'Bryant, Which Fibonacci numbers are the sum of two squares?, MathOverflow.

Crossrefs

Cf. A000045, A001481, A124132.

Programs

Mathematica

Reap[For[n = 0, n <= 400, n = n+2, If[Reduce[Fibonacci[n] == x^2 + y^2, {x, y}, Integers] =!= False, Print[n]; Sow[n]]]][[2, 1]]

PARI

is(n)=if(n%2, return(0)); my(f=factor(fibonacci(n))); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 21 2014

PARI

default(factor_add_primes, 1);
is(n)={
    if(n%2,return(0));
    my(f=fibonacci(n),t);
    if(f%4==3,return(0));
    forprime(p=2,min(log(f)^2,1e5),
        if(f%p==0,
            t=valuation(f,p);
            if(p%4==3&&t%2,return(0));
            f/=p^t;
            if(f%4==3,return(0))
        )
    );
    fordiv(n,d,
        if(d==n, break);
        t=factor(fibonacci(d))[,1];
        for(i=1,#t,
            if(t[i]%4==3 && valuation(f,t[i])%2, return(0));
            f/=t[i]^valuation(f,t[i]);
            if(f%4==3,return(0))
        )
    );
    f=factor(f);
    for(i=1,#f[,1],
        if(f[i,2]%2&&f[i,1]%4==3,return(0))
    );
    1
}; \\ Charles R Greathouse IV, Jan 21 2014

Python

from itertools import count, islice
from sympy import factorint, fibonacci
def A236264_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(0,2))
a236264_list = list(islice(A236264_gen(),10)) # Chai Wah Wu, Jun 27 2022

Formula

a(n) = 2*A124132(n-1).

Extensions

a(32)-a(42) from Charles R Greathouse IV, Jan 21 2014

a(43) from Chai Wah Wu, Jul 23 2020