# HG changeset patch # User Shlomi Fish # Date 1379074847 -10800 # Node ID 9fa99c85adce2b08b7ae9aab4c31af8b853085e3 # Parent 8c8daec2e28bad2cf494f083d49b952e63d9f971 Add the analysis as HTML/LaTeX Notation. diff --git a/project-euler/137/html-analysis.html b/project-euler/137/html-analysis.html new file mode 100644 --- /dev/null +++ b/project-euler/137/html-analysis.html @@ -0,0 +1,86 @@ + + + + +Project Euler #137: Fibonacci Golden Nuggets + + + + + + +

# Project Euler #137: Fibonacci Golden Nuggets

+ + + + +

+$+ +s = \sqrt{5} \\ + +F[n] = \frac{1}{s} \cdot \left\{ \left[ \frac{1+s}{2} \right]^n - \left[\frac{1 - s}{2}\right]^n \right\} + +$ +

+ +Therefore: + +[ Given limits according to the |q| < 1 ] +

+ +

+$+ +A_F(x) = \frac{1}{s} \cdot \left\{ \dfrac{ \frac{x(1+s)}{2} }{ 1 - \frac{x(1+s)}{2} } - \dfrac{ \frac{x(1-s)}{2} }{ 1 - \frac{x(1-s)}{2} } \right\} = \\ + +\frac{x}{2s} \cdot \left\{ \dfrac{ (1+s) }{ 1 - \frac{x(1+s)}{2} } - \dfrac{ (1-s) }{ 1 - \frac{x(1-s)}{2} } \right\} = \\ + +\frac{x}{2s} \cdot \left\{ \dfrac{ 1 }{ \frac{1}{1+s} - x/2 } - \dfrac{ 1 }{ \frac{1}{1-s} - x/2 } \right\} = \\ + + \left[y = x/2\right] \\ + +ys \cdot \left\{ \dfrac{ 1 }{ \frac{1}{1+s} - y } - \dfrac{ 1 }{ \frac{1}{1-s} - y } \right\} = \\ + + \left[z = 1/y ; \cdot \dfrac{\frac{1}{y}}{\frac{1}{y}} \right] \\ + +s \cdot \left\{ \dfrac{1}{\frac{z}{1+s} - 1} - \dfrac{1}{\frac{z}{1-s} - 1} \right\} = N \\ + +s \cdot \dfrac{ \left[\frac{z}{1-s}-1 - \frac{z}{1+s} + 1 \right]}{[\frac{z}{1+s}-1][\frac{z}{1-s}-1]} = N \\ + +sz \cdot (\frac{1}{1-s}-\frac{1}{1+s}) = N\left[\frac{z}{1+s}-1\right]\left[\frac{z}{1-s}-1\right] \\ + + \text{[ * (1+s) * (1-s)]} \\ + + sz(1+s-1+s) = N\left[z-(1+s)\right]\left[z-(1-s)\right] \\ + + 2 s^2 \cdot z = N \left[ z^2 - 2z + (1-s^2)\right] \\ + + s^2 = 5 \\ + + 10z = N \left[ z^2 - 2z - 4 \right] \\ + + z^2 - (2+10/N)z - 4 = 0 \\ + + z = \dfrac{ -b \pm \sqrt{b^2-4ac}}{2a} \\ + + \text{z is rational iff b^2-4ac is a square of a rational number:} \\ + + ∆ = (2+10/N)^2+16 = 4\left[ (1+5/N)^2+4 \right] = 4[ (N+5)^2/N^2 + 4 ] \\ + + ∆ = 4\dfrac{ (N+5)^2+4N^2 }{ N^2 } \\ + + \text{rational iff (N+5)^2+4N^2 is a whole square.} + +$ +

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