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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Fibonacci numbers as coefficients in three variable system of equations

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