## Retractions B

### Problem 446

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 180; Difficulty rating: 60%
For every integer n>1, the family of functions f_{n,a,b} is defined
by f_{n,a,b}(`x`)≡a`x`+b mod n for a,b,`x` integer and 0<a<n, 0≤b<n, 0≤`x`<n.

We will call f_{n,a,b} a *retraction* if f_{n,a,b}(f_{n,a,b}(`x`))≡f_{n,a,b}(`x`) mod n for every 0≤`x`<n.

Let R(n) be the number of retractions for n.

F(N)=∑R(n^{4}+4) for 1≤n≤N.

F(1024)=77532377300600.

Find F(10^{7}) (mod 1 000 000 007)