Fibonacci numbers as coefficients in three variable system of equations

[fibonacci235]

I've been fooling around with the Fibonacci sequence for some time and I thought it would be interesting to see what would happen if I made the coefficients in a three variable system of equations follow that sequence. The results are intriguing and enigmatic.

I discovered the following:

Consider some general homogeneous three variable system of equations:
a11x1 + a12x2 + a13x3 = 0
a21x1 + a22x2 + a23x3 = 0
a31x1 + a32x2 + a33x3 = 0

it happens that if you choose some Fibonacci number Fn such that n ≤ 12 for the value of a11 and then make a12 the next Fibonacci number and a13 the next and a21 the next and so forth, that the system has no solutions...ever and I can't figure out why.

The system of equations begins to take on solutions when a11=233 or n=13 since 233 is the 13th Fibonacci number. In the case of a11=233, the solution is (0,0,0). I have yet to explore how a three variable system of equations behave in a non-homogenous system using the same Fibonacci numbers as coefficients. In the meantime, if anyone can explain why the first 12 Fibonacci numbers, if used in order and all are positive and are used as coefficients in a homogenous three variable system of equations create no solution, I would be eternally grateful.

 

[micromass]

I don't really get it. Isn't (0,0,0) always a solution?? Or did you mean nonzero solutions??

Whether nonzero solutions exists depends on the determinant of the matrix.

 

[fibonacci235]

Ok, I need to apologize. I wasn't being careful enough, and I discovered that I was inputting a wrong number into the computer giving me inaccurate results. However, I still managed to find some interesting things after the correction.

Ignore what I said about the (0,0,0) solution. That is incorrect. I found that if you allow a11=Fn:n<= 12 that the system of equations has no solution. If you allow a11=233 or greater i.e. Fn:n=14,15,16,17,...p the solution is always infinite. This is the case for homogeneous systems at least.

So if you start an augmented matrix like:

| 0 1 1 |= 0
| 2 3 5 |= 0
| 8 13 21 |= 0

here a11=0 which is Fsub1. Next shift the elements so that a11=1 which is Fsub2 etc etc. Keep doing this until a11= 144 which is Fsub12.

You will find that the system never produces solutions. Then when you get to Fsub13= 233 i.e.

| 233 377 610 |=0
| 987 1597 2584 |=0
| 4181 6765 10946 |=0 the solution is always infinity. The same holds true for Fsub14...Fsubn. Making a11=377, 610, 987...n.

I think this is interesting. I created an arbitrary matrix with elements greater than the elements in the augmented matrix produced by Fsub12 and my answer was zero.

I'm sure the determinant has a lot to do with why I'm either getting no solution or infinite solutions, but I still don't understand why. I just thought this was a mathematical curiosity that should be investigated.