How Can I Solve a Modulo Equation with Large Values?

[mathnub31]

How would one normally solve this type of equation

x^a = b (mod n)

Is there any trick to solve it if I know that n = 465992738619896000 and a = 23407534262244700, or perhaps an algorithm?

 

[Eynstone]

The existence can be checked with the help of reciprocity laws, but it's a formidable task ( to speak nothing about HOW to find x).

 

[mathwonk]

i.e, google quadratic reciprocity, for the case of a = 2.

 

[Xitami]

http://www.alpertron.com.ar/LOGDI.HTM

 

[blue_raver22]

Solving modulo equations involves finding the value of x that satisfies the given equation in modular arithmetic. To solve an equation of the form x^a = b (mod n), one would typically use a technique called "exponentiation by squaring." This involves breaking down the exponent a into its binary representation and using repeated squaring to simplify the calculation.

In the case of n = 465992738619896000 and a = 23407534262244700, we can use the following algorithm to solve the equation:

1. Write a in binary form: a = (1011010110101010011001010110001100111100100100)

2. Start with x = 1 and b = b (mod n).

3. For each bit in a (starting from the right), if the bit is 1, multiply x by itself and then multiply by b (mod n). If the bit is 0, simply square x.

4. After going through all the bits in a, the final value of x will be the solution to the equation x^a = b (mod n).

Using this algorithm, we can solve the equation x^23407534262244700 = b (mod 465992738619896000) in a relatively efficient manner. However, it should be noted that solving this equation for such large values of n and a may still be a computationally intensive task.