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

1,1

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, ...

Links

Table of n, a(n) for n=1..37.

Example

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.

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):

Crossrefs

Cf. A000045, A007598, A244855.

Sequence in context: A066844 A255608 A309623 * A366428 A195564 A147287

Adjacent sequences: A242071 A242072 A242073 * A242075 A242076 A242077

Keyword

nonn

Author

Michel Lagneau, Aug 14 2014

Status

approved