A242074 Numbers n such that n^2 - 1 is the product of four distinct Fibonacci numbers greater than 1.

25, 41, 64, 103, 131, 169, 271, 274, 281, 441, 713, 901, 1156, 1871, 3025, 4894, 7921, 12817, 20736, 21319, 33551, 54289, 87842, 142129, 229969, 372100, 602071, 974169, 1576238, 2550409, 4126649, 6677056, 10803703, 17480761, 28284466, 45765225, 74049689

Offset: 1

Author

Michel Lagneau, Aug 14 2014

Keywords

• nonn

Comments

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

Proof:

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

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.)

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;

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).

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

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

Examples

			25^2 - 1 = 2*3*8*13 = F(5 - 2)*F(5 - 1)*F(5 + 1)*F(5 + 2) where F(5) = 5;
41^2 - 1 = 2*5*8*21;
64^2 - 1 = 3*5*13*21 = F(6 - 2)*F(6 - 1)*F(6 + 1)*F(6 + 2) where F(6) = 8;
103^2 - 1 = 3*8*13*34;
131^2 - 1 = 3*8*13*55;
169^2 - 1 = 5*8*21*34 = F(7 - 2)*F(7 - 1)*F(7 + 1)*F(7 + 2) where F(7) = 13;
271^2 - 1 = 3*5*34*144;
274^2 - 1 = 5*13*21*55;
281^2 - 1 = 2*5*8*987;
441^2 - 1 = 8*13*34*55 = F(8 - 2)*F(8 - 1)*F(8 + 1)*F(8 + 2) where F(8) = 21.
		

Crossrefs

Cf. A000045, A007598, A244855.

Programs

Maple

with(combinat,fibonacci):with(numtheory):nn:=100:lst:={}:T:=array(1..nn):
   for n from 1 to nn do:
    T[n]:=fibonacci(n):
   od:
     for p from 1 to nn-1 do:
       for q from p+1 to nn-1 do:
          for r from q+1 to nn-1 do:
            for s from r+1 to nn-1 do:
           f:=T[p]*T[q]*T[r]*T[s]+1:x:=sqrt(f):
           if x=floor(x)and T[p]<>1
           then
           lst:=lst union {x}:
           else
           fi:
          od:
       od:
     od:
    od:
     print(lst):