## Modular inverses

### Problem 451

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}