Commits

Shlomi Fish committed 98b06fe

Remove the old and deprecated analysis.txt.

Comments (0)

Files changed (1)

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.
-
-
-
Tip: Filter by directory path e.g. /media app.js to search for public/media/app.js.
Tip: Use camelCasing e.g. ProjME to search for ProjectModifiedEvent.java.
Tip: Filter by extension type e.g. /repo .js to search for all .js files in the /repo directory.
Tip: Separate your search with spaces e.g. /ssh pom.xml to search for src/ssh/pom.xml.
Tip: Use ↑ and ↓ arrow keys to navigate and return to view the file.
Tip: You can also navigate files with Ctrl+j (next) and Ctrl+k (previous) and view the file with Ctrl+o.
Tip: You can also navigate files with Alt+j (next) and Alt+k (previous) and view the file with Alt+o.