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

1,2

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

Links

Table of n, a(n) for n=1..43.

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.

Formula

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

Example

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

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]]

Prog

(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

Crossrefs

Cf. A000045, A001481, A124132.

Sequence in context: A161922 A027863 A261978 * A152301 A152389 A114103

Adjacent sequences: A236261 A236262 A236263 * A236265 A236266 A236267

Keyword

nonn,more

Author

Jean-François Alcover, Jan 21 2014

Extensions

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

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

Status

approved