How Can I Solve a Modulo Equation with Large Values?

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Discussion Overview

The discussion centers on solving the modulo equation x^a = b (mod n), particularly with large values for n and a. Participants explore methods and algorithms that may assist in finding solutions or verifying existence.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about methods or tricks for solving the equation given large values of n and a.
  • Another participant suggests that the existence of a solution can be checked using reciprocity laws, noting the complexity of the task.
  • A further comment references quadratic reciprocity specifically for the case when a = 2 as a potential approach.
  • A link is provided to a resource that may offer additional insights or tools related to the problem.

Areas of Agreement / Disagreement

Participants express differing views on the methods available for solving the equation, indicating that multiple approaches may exist without a consensus on the best method.

Contextual Notes

The discussion does not resolve the specific mathematical steps required to find x, nor does it clarify the assumptions underlying the use of reciprocity laws or the referenced resource.

mathnub31
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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?
 
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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).
 
i.e, google quadratic reciprocity, for the case of a = 2.
 

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