Math 2.0

Learn Year 1

Elementary number theory

178 ideas in teaching order. Each idea comes after the ideas that it builds on.

Start with Divisibility →

  1. Divisibility definition
  2. Division algorithm statement
  3. Modulo operation definition
  4. Greatest common divisor definition
  5. Euclidean algorithm concept
  6. Bézout's identity statement
  7. Extended Euclidean algorithm concept
  8. Coprime integers definition
  9. Least common multiple definition
  10. Product of the gcd and the lcm statement
  11. Lcm of coprime integers statement
  12. Linear Diophantine equation concept
  13. Prime number definition
  14. Composite number definition
  15. Euclid's lemma statement
  16. Generalized Euclid's lemma statement
  17. Existence of a prime divisor statement
  18. Smallest prime factor definition
  19. Prime factorization definition
  20. Existence of prime factorization statement
  21. Fundamental theorem of arithmetic statement
  22. P-adic valuation definition
  23. P-adic valuation of a product statement
  24. Square-free integer definition
  25. Every integer is a square times a square-free integer statement
  26. Euclid's theorem statement
  27. A composite number has a prime factor at most its square root statement
  28. Trial division concept
  29. Sieve of Eratosthenes concept
  30. Prime-counting function definition
  31. Divergence of the sum of the reciprocals of the primes statement
  32. Bertrand's postulate statement
  33. Prime number theorem statement
  34. Dirichlet's theorem on arithmetic progressions statement
  35. Infinitely many primes congruent to 1 modulo n statement
  36. Euler product formula statement
  37. Twin prime conjecture conjecture
  38. Goldbach's conjecture conjecture
  39. Congruence modulo n definition
  40. Congruence is compatible with addition statement
  41. Congruence is compatible with multiplication statement
  42. Congruence is compatible with powers statement
  43. Cancellation in congruences statement
  44. Residue class definition
  45. Complete residue system definition
  46. Reduced residue system definition
  47. Integers modulo n definition
  48. Integers modulo a prime form a field statement
  49. Divisibility rule concept
  50. Divisibility by 3 statement
  51. Divisibility by 9 statement
  52. Divisibility by 11 statement
  53. Casting out nines concept
  54. Modular multiplicative inverse definition
  55. Existence of inverses modulo n statement
  56. Linear congruence concept
  57. Chinese remainder theorem statement
  58. Chinese remainder theorem for non-coprime moduli statement
  59. Fermat's little theorem statement
  60. Fermat's little theorem in the form a^p = a statement
  61. Euler's totient function definition
  62. Totient of a prime statement
  63. Totient of a prime power statement
  64. Multiplicativity of the totient function statement
  65. Number of invertible residues modulo n statement
  66. Sum of the totient over the divisors statement
  67. Euler's theorem statement
  68. Wilson's theorem statement
  69. Converse of Wilson's theorem statement
  70. Fermat pseudoprime definition
  71. Fermat primality test concept
  72. Carmichael number definition
  73. Modular exponentiation concept
  74. Multiplicative order definition
  75. The order modulo p divides p − 1 statement
  76. Primitive root modulo n definition
  77. Existence of primitive roots modulo a prime statement
  78. Primitive root theorem statement
  79. Discrete logarithm definition
  80. Legendre's formula statement
  81. Kummer's theorem statement
  82. Lucas's theorem statement
  83. Lifting-the-exponent lemma statement
  84. Hensel's lemma statement
  85. Quadratic residue definition
  86. Legendre symbol definition
  87. Euler's criterion statement
  88. Multiplicativity of the Legendre symbol statement
  89. First supplement to quadratic reciprocity statement
  90. −1 is a square modulo p if and only if p is not 3 modulo 4 statement
  91. Second supplement to quadratic reciprocity statement
  92. Gauss's lemma on quadratic residues statement
  93. Eisenstein's lemma statement
  94. Law of quadratic reciprocity statement
  95. Jacobi symbol definition
  96. Quadratic reciprocity for the Jacobi symbol statement
  97. Arithmetic function definition
  98. Multiplicative function definition
  99. Completely multiplicative function definition
  100. Divisors of a natural number definition
  101. Divisor function definition
  102. Number of divisors definition
  103. Sum of divisors definition
  104. Multiplicativity of the divisor function statement
  105. Number of distinct prime factors definition
  106. Number of prime factors counted with multiplicity definition
  107. Möbius function definition
  108. Multiplicativity of the Möbius function statement
  109. Dirichlet convolution definition
  110. Dirichlet convolution formula statement
  111. Dirichlet convolution of multiplicative functions is multiplicative statement
  112. The Möbius function is the inverse of the constant function 1 statement
  113. Möbius inversion formula statement
  114. Liouville function definition
  115. von Mangoldt function definition
  116. Perfect number definition
  117. Abundant number definition
  118. Deficient number definition
  119. Mersenne number definition
  120. Mersenne prime definition
  121. If 2^p − 1 is prime then p is prime statement
  122. Euclid–Euler theorem statement
  123. Lucas–Lehmer test statement
  124. Fermat number definition
  125. Fermat prime definition
  126. If 2^n + 1 is prime then n is a power of 2 statement
  127. Fermat numbers are pairwise coprime statement
  128. Pépin's test statement
  129. Diophantine equation concept
  130. Frobenius coin problem statement
  131. Pythagorean triple definition
  132. Classification of Pythagorean triples statement
  133. Proof by infinite descent concept
  134. Fermat's right triangle theorem statement
  135. Fermat's Last Theorem statement
  136. Fermat's Last Theorem for n = 4 statement
  137. Fermat's Last Theorem for n = 3 statement
  138. Fermat's theorem on sums of two squares statement
  139. Brahmagupta–Fibonacci identity statement
  140. Sum of two squares theorem statement
  141. Euler's four-square identity statement
  142. Lagrange's four-square theorem statement
  143. Legendre's three-square theorem statement
  144. Pell's equation definition
  145. Existence of nontrivial solutions of Pell's equation statement
  146. Fundamental solution of Pell's equation definition
  147. Solutions of Pell's equation are powers of the fundamental solution statement
  148. Minkowski's theorem statement
  149. Chevalley–Warning theorem statement
  150. Continued fraction definition
  151. Simple continued fraction definition
  152. Continued fraction expansion of a real number definition
  153. Convergents of a continued fraction definition
  154. The continued fraction of a rational number terminates statement
  155. Convergents converge to the number statement
  156. Periodic continued fraction definition
  157. Dirichlet's approximation theorem statement
  158. Legendre's theorem on continued fractions statement
  159. Farey sequence definition
  160. Irrationality of square roots of non-squares statement
  161. Algebraic number definition
  162. Transcendental number definition
  163. Liouville number definition
  164. Liouville's theorem on Diophantine approximation statement
  165. Lindemann–Weierstrass theorem statement
  166. Gaussian integer definition
  167. Norm of a Gaussian integer definition
  168. The Gaussian integers form a Euclidean domain statement
  169. Gaussian prime definition
  170. A prime is a Gaussian prime if and only if it is 3 modulo 4 statement
  171. Quadratic integer definition
  172. Consecutive Fibonacci numbers are coprime statement
  173. Bernoulli number definition
  174. Primality test concept
  175. Miller–Rabin primality test concept
  176. Public-key cryptography concept
  177. RSA cryptosystem example
  178. Diffie–Hellman key exchange example