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Numbers n such that n^2 - 1 is the product of four distinct Fibonacci numbers greater than 1.
1

%I #15 Dec 11 2015 13:13:37

%S 25,41,64,103,131,169,271,274,281,441,713,901,1156,1871,3025,4894,

%T 7921,12817,20736,21319,33551,54289,87842,142129,229969,372100,602071,

%U 974169,1576238,2550409,4126649,6677056,10803703,17480761,28284466,45765225,74049689

%N Numbers n such that n^2 - 1 is the product of four distinct Fibonacci numbers greater than 1.

%C The sequence contains the squares of the Fibonacci numbers (A007598(n) for n >=5).

%C Proof:

%C Let F(m) be the m-th Fibonacci number. If n = F(m)^2, n^2 - 1 = F(m)^4-1.

%C For m > 1, F(m)^4 - 1 = F(m-2)*F(m-1)*F(m+1)*F(m+2) with the property F(m-2) + F(m-1) + F(m+1) + F(m+2) = F(m) + F(m+3) = 2*F(m+2). (See A244855.)

%C F(m)^2 - 1 = F(m-1)*F(m+1) if m odd, and F(m)^2 - 1 = F(m-2)*F(m+2)if m even;

%C F(m)^2 + 1 = F(m-2)*F(m+2) if m odd, and F(m)^2 + 1 = F(m-1)*F(m+1) if m even, hence the product (F(m)^2 - 1)*(F(m)^2 + 1) = F(m-2)*F(m-1)*F(m+1)*F(m+2).

%C The primes of the sequence are 41, 103, 131, 271, 281, 1871, 21319, ...

%C The composites (nonsquares) of the sequence are 274, 713, 901, 4894, 12817, 33551, 87842, ...

%e 25^2 - 1 = 2*3*8*13 = F(5 - 2)*F(5 - 1)*F(5 + 1)*F(5 + 2) where F(5) = 5;

%e 41^2 - 1 = 2*5*8*21;

%e 64^2 - 1 = 3*5*13*21 = F(6 - 2)*F(6 - 1)*F(6 + 1)*F(6 + 2) where F(6) = 8;

%e 103^2 - 1 = 3*8*13*34;

%e 131^2 - 1 = 3*8*13*55;

%e 169^2 - 1 = 5*8*21*34 = F(7 - 2)*F(7 - 1)*F(7 + 1)*F(7 + 2) where F(7) = 13;

%e 271^2 - 1 = 3*5*34*144;

%e 274^2 - 1 = 5*13*21*55;

%e 281^2 - 1 = 2*5*8*987;

%e 441^2 - 1 = 8*13*34*55 = F(8 - 2)*F(8 - 1)*F(8 + 1)*F(8 + 2) where F(8) = 21.

%p with(combinat,fibonacci):with(numtheory):nn:=100:lst:={}:T:=array(1..nn):

%p for n from 1 to nn do:

%p T[n]:=fibonacci(n):

%p od:

%p for p from 1 to nn-1 do:

%p for q from p+1 to nn-1 do:

%p for r from q+1 to nn-1 do:

%p for s from r+1 to nn-1 do:

%p f:=T[p]*T[q]*T[r]*T[s]+1:x:=sqrt(f):

%p if x=floor(x)and T[p]<>1

%p then

%p lst:=lst union {x}:

%p else

%p fi:

%p od:

%p od:

%p od:

%p od:

%p print(lst):

%Y Cf. A000045, A007598, A244855.

%K nonn

%O 1,1

%A _Michel Lagneau_, Aug 14 2014