UsefulLinks
Mathematics
Number Theory
1. Foundations of Number Theory
2. Divisibility Theory
3. Prime Numbers and Factorization
4. Greatest Common Divisor and Least Common Multiple
5. Modular Arithmetic and Congruences
6. Classical Theorems in Number Theory
7. Arithmetic Functions
8. Diophantine Equations
9. Quadratic Residues and Reciprocity
10. Prime Number Theory
11. Continued Fractions
12. Elementary Analytic Number Theory
13. Cryptographic Applications
14. Advanced Topics and Connections
9.
Quadratic Residues and Reciprocity
9.1.
Quadratic Residues
9.1.1.
Definition and Basic Properties
9.1.2.
Quadratic Non-residues
9.1.3.
Examples and Computations
9.2.
Legendre Symbol
9.2.1.
Definition and Notation
9.2.2.
Basic Properties
9.2.3.
Computational Rules
9.2.4.
Multiplicativity
9.3.
Euler's Criterion
9.3.1.
Statement and Proof
9.3.2.
Connection to Fermat's Little Theorem
9.4.
Gauss's Lemma
9.4.1.
Statement and Proof
9.4.2.
Applications to Quadratic Reciprocity
9.5.
Quadratic Reciprocity Law
9.5.1.
Main Reciprocity Law
9.5.2.
First Supplement
9.5.2.1.
Quadratic Character of -1
9.5.3.
Second Supplement
9.5.3.1.
Quadratic Character of 2
9.5.4.
Proof Strategies
9.5.5.
Applications and Examples
9.6.
Jacobi Symbol
9.6.1.
Definition and Properties
9.6.2.
Relationship to Legendre Symbol
9.6.3.
Computational Advantages
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8. Diophantine Equations
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10. Prime Number Theory