A245046 Smallest non-Fibonacci number k such that k^2 + F(n)^2 = f1*f2 where F(n) = A000045(n) and f1, f2 are distinct Fibonacci numbers.

70, 70, 6, 991, 27, 183, 443, 38, 27, 373

Offset: 1

Author

Michel Lagneau, Jul 16 2014

Keywords

• nonn
• more

Comments

The sequence is probably finite.

Conjecture : if k exists, the number of solutions non-Fibonacci k of the equation k^2 + F(n)^2 = f1*f2 is finite.

The primes of the sequence are 373, 443 and 991.

If f1<=f2, the new sequence is 70, 70, 6, 4, 12, 183, 443, 38, 27, 373

Examples

			a(1) = a(2) = 70 because 70^2+1 = F(7)*F(14) = 13*377. The number 70 is probably unique.
a(3) = 6 because 6^2+2^2 = F(5)*F(6) = 5*8. But there exists also k = 10 such that 10^2+4 = F(6)*F(7) = 8*13.
a(7) = 443 because 443^2 + 13^2 = F(1)*F(27) = 1*196418.
		

Crossrefs

Cf. A000045, A245236.

Programs

Maple

with(combinat,fibonacci):with(numtheory):nn:=200:T:=array(1..nn):
  for i from 1 to nn do:
    T[i]:=fibonacci(i):
  od:
   for n from 1 to 10 do:
     ff:=fibonacci(n):ii:=0:
       for p from 1 to nn-1 while(ii=0)do:
         for q from p+1 to nn-1 while(ii=0)do:
           f:=T[p]*T[q]-ff^2:x:=sqrt(f):x1:=sqrt(5*f+4):x2:=sqrt(5*f-4):
           if f>0 and x=floor(x)
           and x1<>floor(x1) and x2<>floor(x2)
           then
           ii:=1:printf ( "%d %d %d %d \n",n,x,T[p],T[q]):
           else
           fi:
         od:
       od:
    od:

PARI

isfib(n) = {my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));} \\ from A010056
isprod(pf) = {sqrpf = sqrtint(pf); ifib = 1; while((fif = fibonacci(ifib)) < sqrpf, if (pf % fif == 0, if (isfib(pf/fif), return (1));); ifib ++;); return (0);}
a(n) = {k = 1; fsq = fibonacci(n)^2; ok = 0; while (!ok, if (! isfib(k), pf = k^2 + fsq; ok = isprod(pf);); if (! ok, k++);); k;} \\ Michel Marcus, Jul 17 2014