Thought Process

First thing to notice is that 13082761331670030 = 2*3*5*...*43

This means that solving for x^3 = 1 mod(13082761331670030) is equivalent to solving the following system of equations by the Chinese Remainder Theorem

1. x^3 = 1 mod(2)

2. x^3 = 1 mod(3)

3. ...

4. x^3 = 1 mod(43)

So first I need to write a Chinese Remainder Function (CRF), which is now included in my Essential functions page!

Then we find the solutions for all of these and continuously build solutions, for example:

1. N = 91 = 7*13, we must solve

2. x^3 = 1 mod(7)

   1. There are 3 solutions: 1, 2, 4

3. x^3 = 1 mod(13)

   1. There are 3 solutions: 1, 3, 9

4. Now I build my solutions with CRF, which takes 4 inputs a1, a2, n1, n2 where x = a1 mod n1, x = a2 mod n2, and finds the unique solutions

5. CRF(1, 1, 7, 13) = 1

6. CRF(1, 3, 7, 13) = 29

7. CRF(1, 9, 7, 13) = 22

8. CRF(2, 1, 7, 13) = 79

9. CRF(2, 3, 7, 13) = 9

10. CRF(2, 9, 7, 13) = 16

11. CRF(4, 1, 7, 13) = 53

12. CRF(4, 3, 7, 13) = 81

13. CRF(4, 9, 7, 13) = 74

This gives us all the solutions!