Metamath Proof Explorer


Table of Contents - 14.4.8. Quadratic residues and the Legendre symbol

If the congruence has a solution we say that is a quadratic residue. If the congruence has no solution we say that is a quadratic nonresidue, see definition in [ApostolNT] p. 178. The Legendre symbol is defined in a way that its value is if is a quadratic residue and if is a quadratic nonresidue (and if divides ), see lgsqr.

Originally, the Legendre symbol was defined for odd primes only (and arbitrary integers ) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs for the "Legendre symbol" is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for , but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol".

  1. clgs
  2. df-lgs
  3. zabsle1
  4. lgslem1
  5. lgslem2
  6. lgslem3
  7. lgslem4
  8. lgsval
  9. lgsfval
  10. lgsfcl2
  11. lgscllem
  12. lgsfcl
  13. lgsfle1
  14. lgsval2lem
  15. lgsval4lem
  16. lgscl2
  17. lgs0
  18. lgscl
  19. lgsle1
  20. lgsval2
  21. lgs2
  22. lgsval3
  23. lgsvalmod
  24. lgsval4
  25. lgsfcl3
  26. lgsval4a
  27. lgscl1
  28. lgsneg
  29. lgsneg1
  30. lgsmod
  31. lgsdilem
  32. lgsdir2lem1
  33. lgsdir2lem2
  34. lgsdir2lem3
  35. lgsdir2lem4
  36. lgsdir2lem5
  37. lgsdir2
  38. lgsdirprm
  39. lgsdir
  40. lgsdilem2
  41. lgsdi
  42. lgsne0
  43. lgsabs1
  44. lgssq
  45. lgssq2
  46. lgsprme0
  47. 1lgs
  48. lgs1
  49. lgsmodeq
  50. lgsmulsqcoprm
  51. lgsdirnn0
  52. lgsdinn0
  53. lgsqrlem1
  54. lgsqrlem2
  55. lgsqrlem3
  56. lgsqrlem4
  57. lgsqrlem5
  58. lgsqr
  59. lgsqrmod
  60. lgsqrmodndvds
  61. lgsdchrval
  62. lgsdchr