# Commits

committed 98b06fe

Remove the old and deprecated analysis.txt.

# project-euler/137/analysis.txt

`-F[n] = 1/sqrt(5) * { [ (1 + sqrt(5)) / 2 ]^n - [ (1 - sqrt(5)) / 2 ] ^ n }`
`-`
`-\$s = sqrt(5)`
`-`
`-Therefore:`
`-`
`-[ Given limits according to the |q| <= 1 ]`
`-`
`-A_F(x) = 1/\$s * { x(1+\$s)/2/[ 1 - x(1+\$s)/2] - x(1-\$s)/2/[1 - x(1-\$s)/2] } =`
`-`
`-x/(2*\$s) * { (1+\$s)/[1 - x(1+\$s)/2] - (1-\$s)/[1 - x(1-\$s)/2] } =`
`-`
`-x/(2*\$s) * { 1/[1/(1+\$s) - x/2] - 1/[1/(1-\$s) - x/2] } =`
`-`
`-    [y = x/2]`
`-`
`-y*\$s * {1/[1/(1+\$s) - y] - 1/[1/(1-\$s) - y] } =`
`-`
`-    [z = 1/y ; * (1/y)/(1/y) ]`
`-`
`-\$s * { 1 / [z/(1+\$s) - 1] - 1 / [z/(1-\$s) - 1] } = N`
`-`
`-\$s * [z/(1-\$s)-1 - z/(1+\$s) + 1 ]/[z/(1+s)-1]/[z/(1-s)-1] = N`
`-`
`-\$s * z * (1/(1-\$s)-1/(1+\$s)) = N[z/(1+\$s)-1][z/(1-s)-1]`
`-`
`-    [ * (1+\$s) * (1-\$s)]`
`-`
`-\$s * \$z * (1+\$s-1+\$s) = N[z-(1+\$s)][z-(1-\$s)]`
`-`
`-    2 \$s^2 * \$z = N [ z^2 - 2z + (1-\$s^2)]`
`-`
`-    \$s^2 = 5`
`-`
`-    10z = N [ z^2 - 2z - 4 ]`
`-`
`-    z^2 - (2+10/N)z - 4 = 0`
`-`
`-    z = [ -b +/- sqrt(b^2-4ac) ] / (2*a)`
`-`
`-    z is rational iff b^2-4ac is a square of a rational number:`
`-`
`-    ∆ = (2+10/N)^2+16 = 4[ (1+5/N)^2+4 ] = 4[ (N+5)^2/N^2 + 4 ]`
`-`
`-    ∆ = 4[ [ (N+5)^2+4N^2 ] / N^2 ]`
`-`
`-        rational iff (N+5)^2+4N^2 is a whole square.`
`-`
`-`
`-`
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