Number theory - quadratic residues

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SUMMARY

The discussion focuses on solving quadratic congruences, specifically finding incongruent solutions for the equations X² ≡ 23 (mod 77) and X² ≡ 11 (mod 39). Participants highlight the necessity of applying the Chinese Remainder Theorem to break down the moduli into their prime factors, 7 and 11 for the first equation, and 3 and 13 for the second. The conversation emphasizes the importance of modular arithmetic in number theory and provides a direct link to the relevant Wikipedia page for further reference.

PREREQUISITES
  • Understanding of quadratic congruences
  • Familiarity with modular arithmetic
  • Knowledge of the Chinese Remainder Theorem
  • Basic number theory concepts
NEXT STEPS
  • Study the Chinese Remainder Theorem in detail
  • Learn about solving quadratic congruences
  • Explore modular arithmetic techniques
  • Investigate applications of quadratic residues in cryptography
USEFUL FOR

Students of number theory, mathematicians interested in modular arithmetic, and anyone solving quadratic congruences in mathematical research or coursework.

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number theory -- quadratic residues

Homework Statement



find all incongruent solutions of each quadratic congruence below.
X^2\equiv23 mod 77

Homework Equations


X^2\equiv11 mod 39


The Attempt at a Solution


it is suffices to X^2\equiv23 mod 7, andX^2\equiv23 mod 11,
then how to do next?
 
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