## Modular inverses

### Problem 451

Published on Sunday, 22nd December 2013, 10:00 am; Solved by 578; Difficulty rating: 30%
Consider the number 15.

There are eight positive numbers less than 15 which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14.

The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14

because

1*1 mod 15=1

2*8=16 mod 15=1

4*4=16 mod 15=1

7*13=91 mod 15=1

11*11=121 mod 15=1

14*14=196 mod 15=1

Let I(n) be the largest positive number m smaller than n-1 such that the modular inverse of m modulo n equals m itself.

So I(15)=11.

Also I(100)=51 and I(7)=1.

Find ∑I(n) for 3≤n≤2·10^{7}