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A236264 Even indices of Fibonacci numbers which are the sum of two squares. 0
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 (list; graph; refs; listen; history; text; internal format)
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, 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

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-Fran├žois 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

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)