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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.

%I A245046 #31 Jul 20 2014 09:51:09
%S A245046 70,70,6,991,27,183,443,38,27,373
%N 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.
%C A245046 The sequence is probably finite.
%C A245046 Conjecture : if k exists, the number of solutions non-Fibonacci k of the equation k^2 + F(n)^2 = f1*f2 is finite.
%C A245046 The primes of the sequence are 373, 443 and 991.
%C A245046 If f1<=f2, the new sequence is 70, 70, 6, 4, 12, 183, 443, 38, 27, 373
%e A245046 a(1) = a(2) = 70 because 70^2+1 = F(7)*F(14) = 13*377. The number 70 is probably unique.
%e A245046 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.
%e A245046 a(7) = 443 because 443^2 + 13^2 = F(1)*F(27) = 1*196418.
%p A245046 with(combinat,fibonacci):with(numtheory):nn:=200:T:=array(1..nn):
%p A245046   for i from 1 to nn do:
%p A245046     T[i]:=fibonacci(i):
%p A245046   od:
%p A245046    for n from 1 to 10 do:
%p A245046      ff:=fibonacci(n):ii:=0:
%p A245046        for p from 1 to nn-1 while(ii=0)do:
%p A245046          for q from p+1 to nn-1 while(ii=0)do:
%p A245046            f:=T[p]*T[q]-ff^2:x:=sqrt(f):x1:=sqrt(5*f+4):x2:=sqrt(5*f-4):
%p A245046            if f>0 and x=floor(x)
%p A245046            and x1<>floor(x1) and x2<>floor(x2)
%p A245046            then
%p A245046            ii:=1:printf ( "%d %d %d %d \n",n,x,T[p],T[q]):
%p A245046            else
%p A245046            fi:
%p A245046          od:
%p A245046        od:
%p A245046     od:
%o A245046 (PARI) isfib(n) = {my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));} \\ from A010056
%o A245046 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);}
%o A245046 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
%Y A245046 Cf. A000045, A245236.
%K A245046 nonn,more
%O A245046 1,1
%A A245046 _Michel Lagneau_, Jul 16 2014