Here's a new math puzzle

[wpolly]

The sum is pi/2, isn't it?

[hyh1048576]

Originally Posted by wpolly

Let Ui be a Lucas Sequence with Discriment d, and p is a prime.

Prove that p|Up+(d|p). ((d|p) is the Legendre Symbol)

This is just a generalization of the original puzzle.

But I think you have made a mistake: in fact p|Up-(d|p).Here p-(d|p) is the suffix
For example,if Ui is the Fabonacci Sequence,then d=5,we have (5|p)=1 iff p=+-1(mod10),so p|F(p-1).And (5|p)=-1 iff p=+-3(mod10),so p|F(p+1).