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1. What Is Number Theory?
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2. Pythagorean Triples
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3. Pythagorean Triples and the Unit Circle
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4. Sums of Higher Powers and Fermat’s Last Theorem
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5. Divisibility and the Greatest Common Divisor
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6. Linear Equations and the Greatest Common Divisor
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7. Factorization and the Fundamental Theorem of
Arithmetic
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8. Congruences
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9. Congruences, Powers, and Fermat’s Little Theorem
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10. Congruences, Powers, and Euler’s Formula
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11. Euler’s Phi Function and the Chinese Remainder
Theorem
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12. Prime Numbers
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13. Counting Primes
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14. Mersenne Primes
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15. Mersenne Primes and Perfect Numbers
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16. Powers Modulo m and Successive Squaring
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17. Computing kth Roots Modulo m
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18. Powers, Roots, and “Unbreakable” Codes
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19. Primality Testing and Carmichael Numbers
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20. Squares Modulo p
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21. Quadratic Reciprocity
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22. Proof of Quadratic Reciprocity
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23. Which Primes Are Sums of Two Squares?
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24.Which Numbers Are Sums of Two Squares?
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25. Euler’s Phi Function and Sums of Divisors
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26. Powers Modulo p and Primitive Roots
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27. Primitive Roots and Indices
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28. The Equation X4 + Y4 = Z4
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29. Square–Triangular Numbers Revisited
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30. Pell’s Equation
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31. Diophantine Approximation
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32. Diophantine Approximation and Pell’s Equation
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33. Number Theory and Imaginary Numbers
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34. The Gaussian Integers and Unique Factorization
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35. Irrational Numbers and Transcendental Numbers
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36. Binomial Coefficients and Pascal’s Triangle
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37. Fibonacci’s Rabbits and Linear Recurrence Sequences
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38. Cubic Curves and Elliptic Curves
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39. Elliptic Curves with Few Rational Points
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40. Points on Elliptic Curves Modulo p
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41. Torsion Collections Modulo p and Bad Primes
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42. Defect Bounds and Modularity Patterns
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43. Elliptic Curves and Fermat’s Last Theorem
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44. The Topsy-Turvy World of Continud Fractions
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45. Continued Fractions and Pell’s Equation
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46. Generating Functions
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47. Sums of Powers
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48. Appendix: A List of Primes