Irrational jumps
Problem 576
A bouncing point moves counterclockwise along a circle with circumference with jumps of constant length , until it hits a gap of length , that is placed in a distance counterclockwise from the starting point. The gap does not include the starting point, that is .
Let be the sum of the length of all jumps, until the point falls into the gap. It can be shown that is finite for any irrational jump size , regardless of the values of and .
Examples:
, and
.
Examples:
, and
.
Let be the maximum of for all primes and any valid value of .
Examples:
, since is the maximal reachable sum for .
Examples:
, since is the maximal reachable sum for .
Find , rounded to 4 decimal places.