MathDB

Let

\mathcal{C}

be the hyperbola y^2 \minus{} x^2 \equal{} 1. Given a point

P_0

on the

x

-axis, we construct a sequence of points

(P_n)

on the

x

-axis in the following manner: let

\ell_n

be the line with slope

1

passing passing through

P_n

, then P_{n\plus{}1} is the orthogonal projection of the point of intersection of

\ell_n

and

\mathcal C

onto the

x

-axis. (If P_n \equal{} 0, then the sequence simply terminates.) Let

N

be the number of starting positions

P_0

on the

x

-axis such that P_0 \equal{} P_{2008}. Determine the remainder of

N

when divided by

2008

31