Solving congruences and diophantine equations in number theory

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Discussion Overview

The discussion centers on solving systems of congruences and diophantine equations in number theory. Participants explore methods for addressing both first-order and second-order congruences, as well as systems involving multiple unknowns.

Discussion Character

  • Exploratory, Homework-related, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion regarding solving systems of congruences and diophantine equations, seeking examples for clarification.
  • Another participant suggests that providing a specific example of an equation would help in understanding the solution process.
  • A participant presents a specific system of congruences to solve, including both first-order and second-order cases, and requests guidance on applying these methods to similar problems.
  • One participant shares their enjoyment of solving such systems, likening them to a multifaceted gem, indicating a positive engagement with the topic.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants have varying levels of understanding and engagement with the topic. Some seek clarification while others express enjoyment in the complexity of the equations.

Contextual Notes

Participants have not provided specific assumptions or definitions that may affect the understanding of the equations. There are unresolved steps in the solution processes mentioned.

Who May Find This Useful

This discussion may be useful for individuals interested in number theory, particularly those looking to understand congruences and diophantine equations in a mathematical context.

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Hello. I do not understand how to solve systems of three or two congruences of one unknown of first order, a congruence of one unknown of second order and a system of diophantine equations of two or three unknowns. Could someone help me by providing examples in these cases? Thank you.
 
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Why don't you post an example of an equation and what you don't understand about how to solve it
 
Solve the system 8x ≡ 4(mod20), 15x ≡ 10(mod35), 9x ≡ 12(mod39). As well, if you can tell me how to apply it to other similar situations. Find also the solutions of the system x ≡ 1(mod15), x ≡ 7(mod18). Thank you.
 
I enjoy systems of equations such as these. They remind me of a multifaceted gem.
 
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