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The Genesis of Quadratic Reciprocity

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Reciprocity Laws

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

In this first chapter we will present the fathers of the quadratic reciprocity law. Although some results on quadratic residues modulo 10 have been found very early on (see [Ene]) — in connection with the problem of characterizing perfect squares — the history of modern number theory starts with the editions of the books of Diophantus, in particular with the commented edition by Bachet in 1621.

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Additional References

  1. L.M. Adleman, K. Manders, G. Miller, On taking roots in finite fields, Proc. 18th Annual Symp. Computer Science 1977, pp. 175–178

    Google Scholar 

  2. A.G. Agargün, C.R. Fletcher, al-Farisi and the fundamental theorem of arithmetic, Historia Math. 21 (1994), 162–173

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Ahlborn, Uber Berechnung von Summen von größten Ganzen auf geometrischem Wege nach der von Eisenstein zuerst angewandten Methode, Pr. Hamburg (1881), JFM 13. 0141. 04

    Google Scholar 

  4. M. Aigner, G.M. Ziegler, Proofs from THE BOOK, Springer Verlag 1998

    Google Scholar 

  5. N. S. Aladov, Sur la distribution des résidus quadratiques et non-quadratiques d’un nombre premier p dans la suite 1, 2,…,p - 1,Recueil Math. (Moscow Math. Soc.) 18 (1896), 61–75; FdM 27 (1896), 146; FdM 30 (1899), 184; BSMA (2) 21 (1897), 63–64

    Google Scholar 

  6. A. A. Antropov, Partitioning of forms into genera and the reciprocity law in papers of L. Euler, Vopr. Istor. Est. Tekh. (1989), 56–57

    Google Scholar 

  7. A. A. Antropov, On Euler’s partition of forms into genera, Historia Math. 22 (1995), 188–193

    Google Scholar 

  8. E. Artin, Quadratische Körper im Gebiet der höheren Kongruenzen, Math. Z. 19 (1924), 153–246

    Google Scholar 

  9. A. Aubry, L’OEuvre arithmétique d’Euler, L’Ens. Math. 11 (1909), 329–356

    Google Scholar 

  10. A. Aubry, Sur les travaux arithmétiques de Lagrange, de Legendre et de Gauss, L’Ens. Math. 11 (1909), 430–450

    Google Scholar 

  11. E. Bach, A note on square roots in finite fields, IEEE Trans. Inf. Theory 36 (1990), 1494–1498

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Bachmann, Die analytische Zahlentheorie, Leipzig 1894

    Google Scholar 

  13. P. Bachmann, Uber Gauf3’ zahlentheoretische Arbeiten, Abh. Ges. Wiss. Nachr. Gött., Math.-Phys. Kl. 1911, 455–508; FdM 42 (1911), 201–202

    Google Scholar 

  14. R.J. Backlund, Ober die Differenzen zwischen den Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comment. in honorem Ernesti Leonardi Lindelöf, Helsinki 1929

    Google Scholar 

  15. H. Bass, Algebraic K-Theory, New York 1968

    Google Scholar 

  16. A.A. Bennett, Large primes have four consecutive quadratic residues, Tohôku Math. J. 26 (1925), 53–57

    Google Scholar 

  17. K.R. Biermann, E. Schuhmann, H. Wussing, O. Neumann, Mathematisches Tagebuch 1796–1814 von Carl Friedrich Gauss, 3rd edition, Ostwalds Klassiker 256, Leipzig 1981

    Google Scholar 

  18. R.G. Bierstedt, W.H. Mills, On the bound for a pair of consecutive quartic residues of a prime, Proc. Am. Math. Soc. 14 (1963), 628–632

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Brauer, Ober Sequenzen von Potenzresten, Sitzungsbericht Preuß. Akad. Wiss. 1928, 9–26

    Google Scholar 

  20. A. Brauer, H. Zeitz, Ober eine zahlentheoretische Behauptung von Legendre, Sitzber. Berliner Math. Ges. 1930, 116–124; see also A. Brauer, Nachruf auf Hermann Zeitz, ibid. 1933, 2–5

    Google Scholar 

  21. A. Brauer, R.L. Reynolds, On a theorem of Aubry-Thue, Canadian J. Math. 3 (1951), 367–374

    MathSciNet  MATH  Google Scholar 

  22. J. Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), 1011–1013

    Google Scholar 

  23. J. Brillhart, D.H. Lehmer, E. Lehmer, Bound for pairs of consecutive seventh and higher power residues, Math. Comput. 18 (1964), 397–407

    MathSciNet  MATH  Google Scholar 

  24. W.K. Bühler, Gauss: A biographical study, Springer 1981

    Google Scholar 

  25. N.V. Bugaieff, Die Lösung der Congruenzen 2ten Grades in Bezug auf einen Primzahlmodulus, Mosk. Math. Samml. X (1882); FdM 14 (1882), 126–127

    Google Scholar 

  26. Bull K. Burde, Verteilungseigenschaften von Potenzresten, J. Reine Angew. Math. 249 (1971), 133–172

    Google Scholar 

  27. K. Burde, Sequenzen der Länge 2 von Rest klassencharakteren, J. Reine Angew. Math. 272 (1973), 194–202

    Google Scholar 

  28. K. Burde, Ober allgemeine Sequenzen der Länge 3 von Legendre-Symbolen, J. Reine Angew. Math. 282 (1975), 203–216

    Google Scholar 

  29. K. Burde, Eine Verteilungseigenschaft der Legendre-Symbole, J. Number Theory 12 (1980), 273–277

    Google Scholar 

  30. J.J. Burkhardt, Euler’s work on number theory: a concordance for A. Weil’s Number Theory, Historia Math. 13 (1986), 28–35

    Google Scholar 

  31. W.H. Bussey, Fermat’s method of infinite descent, Amer. Math. Monthly 25 (1918), 333–337

    Google Scholar 

  32. P.A. Caris, A solution of the quadratic congruence modulo p, p = 8n+ 1, n odd, Amer. Math. Monthly 32 (1925), 294–297

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Carlitz, Quadratic residues and Tchebycheff polynomials, Portugal. Math. 18 (1959), 193–198

    Google Scholar 

  34. P. Cartier, Sur une generalisation du transfert en theorie des groupes, Enseign. Math. 16 (1970), 49–57

    Google Scholar 

  35. T.-H. Chang, Über aufeinanderfolgende Zahlen, von denen jede mindestens einer von n linearen Kongruenzen genügt, deren Moduln die ersten n Primzahlen sind, Schriften math. Serein. u. Inst. f. angew. Math. Univ. Berlin 4 (1938), Heft 2, 35–55

    Google Scholar 

  36. T.-H. Chang, Lösung der Kongruenz x 2–a (mod p) nach einem Primzahlmodul p = 4n + 1, Math. Nachr. 22 (1960), 136–142

    Google Scholar 

  37. P. Chebyshev, Theorie der Congruenzen (Russ.), 1849; German Transi. 1889

    Google Scholar 

  38. P. Chowla, S. Chowla, Problems on periodic simple continued fractions, Proc. Natl. Acad. S.I. USA 69 (1972), 37–45

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Cipolla, Un metodo per la risoluzione della congruenza di secondo grado, Rend. Accad. Sci. Fis. Mat. Napoli (3) 9 (1903), 153–163

    Google Scholar 

  40. T. Cochrane, P. Mitchell, Small solutions of the Legendre Equation, J. Number Theory 70 (1998), 62–66

    Article  MathSciNet  MATH  Google Scholar 

  41. H. Cohen, A course in computational algebraic number theory, GTM 138, Springer-Verlag 1993

    Google Scholar 

  42. G.E. Collins, R.G.K. Loos, The Jacobi symbol algorithm, SIGSAM Bull. 16 (1982), 12–16

    Article  MATH  Google Scholar 

  43. Ch. Cornaros, On Grzegorczyk induction, Ann,. Pure Appl. Logic 74 (1995), 1–21

    Google Scholar 

  44. J.E. Cremona, Efficient solution of rational conics, preprint 1998 [Cuc] R. Cuculière, Le plus beau théorème de Fermat, Pour la Science, April 1986

    Google Scholar 

  45. H. Davenport, The Higher Arithmetic. An introduction to the theory of numbers, 6th ed Cambridge Univ. Press 1992

    Google Scholar 

  46. H. Davenport, M. Hall, On the equation ax e + by 2 + cz 2 = 0, Quart. J. Math. 19 (1948), 189–192

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Dedekind, Abriss einer Theorie der höheren Kongruenzen in Bezug auf einen reellen Primzahl-Modulus, J. Reine Angew. Math 54 (1857), 1–26; Ges. Math. Werke I, 40–67

    Google Scholar 

  48. P. Delsarte, A generalization of the Legendre symbol for finite abelian groups, Discrete Math. 27 (1979), 187–192

    Article  MathSciNet  MATH  Google Scholar 

  49. L.E. Dickson, History of the Theory of Numbers, vol I, Chelsea Publishing, New York 1952

    Google Scholar 

  50. L.E. Dickson, History of the Theory of Numbers, vol II, Chelsea Publishing, New York 1952

    Google Scholar 

  51. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält, Abh. Preuss. Akad. Wiss. (1837), 45–81; Werke I (1889), 313–342

    Google Scholar 

  52. K. Dörge, Zur Verteilung der quadratischen Reste, Jahresb. DMV 38 (1929), 41–49; FdM 55 I (1929), 95–95

    Google Scholar 

  53. A. Dupré, Examen d’un proposition de Legendre relative à la théorie des nombres, ouvrage placé en premier ligne par l’Academie des Sciences dans le concours pour le grand prix de Mathématiques de 1858, Paris (1859), Mallet-Bachelier

    Google Scholar 

  54. H.M. Edwards, The background of Kummer’s proof of Fermat’s last theorem for regular primes, Arch. Hist. Exact Sci. 14 (1975), 219–236.

    Google Scholar 

  55. H.M. Edwards, Postscript to “The background of Kummer’s proof ”, Arch. Hist. Exact Sci. 18 (1977), 381–394

    Google Scholar 

  56. G. Eneström, Ober die Geschichte der quadratischen und kubischen Reste im christlichen Mittelalter, (Anfrage 140), Bibl. Math. (3), 9 (1908), 265–266

    Google Scholar 

  57. A. Errera, Sur un théorème de Legendre, Bull. Sci. Math. (2) 50 (1926), 166–168

    Google Scholar 

  58. L. Euler, Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia, Novi. Comm. Acad. Sci. Petropol. 18 (1772/74), 85–135; (cf. Opera Omnia vol. 3, pp. 240–281 )

    Google Scholar 

  59. P. Eymard, J.P. Lafon, Le journal mathématique de Gauss, Revue d’histoire des sciences 9 (1956), 21–51

    Article  Google Scholar 

  60. K. Feng, L. Ying, An elementary proof of quadratic reciprocity law in F q [t], J. Sichuan Univ., Nat. Sci. Ed. 26, Spec. Issue (1989) 36–40

    Google Scholar 

  61. W. Fouche, A reciprocity law for polynomials with Bernoulli coefficients, Trans. Am. Math. Soc. 288 (1985), 59–67

    MathSciNet  MATH  Google Scholar 

  62. D.H. Fowler, The mathematics of Plato’s Academy. A new reconstruction, Oxford Univ. Press 1987; revised reprint 1990

    Google Scholar 

  63. C. Friesen, Legendre symbols and continued fractions, Acta Arith. 59 (1991), 365–379

    Google Scholar 

  64. J. Froemke, J.M. Grossman, An algebraic approach to some number-theoretic problems arising from paper-folding regular polygons, Amer. Math. Monthly 95 (1988), 289–307

    Article  MathSciNet  MATH  Google Scholar 

  65. W. Fuchs, Zur Lehre von den Kongruenzen bei C. F. Gauss III, Mitt., Gauss-Ges. Gött. 19 (1982), 63–78

    Google Scholar 

  66. C. F. Gauss, Algorithmus novus ad decidendum, utrum numerus integer positivus datas numeri primi positivi dati residuum quadraticum sit an non-residuum, Werke II, p. 59–68

    Google Scholar 

  67. C. F. Gauss, Anzeige, Göttingische Gelehrte Anzeigen, 12. Mai 1808, Werke II, p. 151–154

    Google Scholar 

  68. J. Gebel, A. Pethö, H.G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), 171–192

    MathSciNet  MATH  Google Scholar 

  69. R.E. Giudici, J. Jtem, On classes of quadratic residues and some applications, Sci., Ser. A 3 (1989), 23–32

    Google Scholar 

  70. C. Goldstein, On a seventeenth-century version of the “fundamental theorem of arithmetic”, Historia Math. 19 (1992), 177–187

    Google Scholar 

  71. C. Goldstein, Un théorème de Fermat et ses lecteurs, Saint-Denis: Presses Universitaires de Vincennes, 1995

    MATH  Google Scholar 

  72. Götting, Uber die Verteilung der Reste und Nichtreste einer Primzahl von der Form 4n + 3 innerhalb des Intervalles 1 bis (p — 1)/2, J. Reine Angew. Math. 70 (1869), 363–364;

    Google Scholar 

  73. J. J. Gray, A commentary on Gauss’s mathematical diary, 1796–1814, with an English translation, Expositiones Math. 2 (1984), 97–130; Erratum: ibid. 3 (1985), 192

    Google Scholar 

  74. L. v. Grosschmid, Zur Theorie der quadratischen Reste,J. Reine Angew. Math. 145 (1915), 254–257; FdM 45 (1914/15), 324

    Google Scholar 

  75. E. Grosswald, Representations of integers as sums of squares, Springer-Verlag 1985

    Google Scholar 

  76. A. Grytczuk, On some connections between Legendre symbols and continued fractions, Acta Acad. Paedagog. Agriensis, Sect. Mat. (N.S.) 24 (1997), 19–21

    Google Scholar 

  77. Gull M. Guinot, Les resveries de Fermat, Aléas, Lyon 1993

    Google Scholar 

  78. M. Guinot, “Le diable d’homme” d’Euler, Aléas, Lyon 1994

    Google Scholar 

  79. M. Guinot, Une époque de transition: Lagrange et Legendre, Aléas, Lyon 1996

    Google Scholar 

  80. H. Gupta, Chains of quadratic residues,Math. Comp. 25 (1971), 379382

    Google Scholar 

  81. Heal T.L. Heath, Diophantus of Alexandria. A study in the history of Greek algebra, Cambridge Univ. Press 1885; Dover Publications, 1964

    Google Scholar 

  82. Y. Hellegouarch, Loi de réciprocité, critère de primalité dans F g [t], C. R. Math. Acad. Sci., Soc. R. Can. 8 (1986), 291–296

    Google Scholar 

  83. Hil D. Hilbert, Natur und mathematisches Erkennen, Lecture Notes, Autumn 1919, Göttingen 1989

    Google Scholar 

  84. J. Hofmann, Uber die zahlentheoretischen Methoden Fermats und Eulers, ihre Zusammenhänge und ihre Bedeutung,Archive for History of Exact Sciences 1 (1960/62), 122–159

    Google Scholar 

  85. L. Holzer, Minimal solutions of diophantine equations, Can. J. Math. 11 (1950), 1–3

    Google Scholar 

  86. H. Hopf, Uber die Verteilung quadratischer Reste, Math. Z. 32 (1930), 222–231

    Google Scholar 

  87. K.R. Johnson, Reciprocity in Ramanujan’s sum, Math. Mag. 59 (1986), 216–222

    Google Scholar 

  88. K. Joshi, Notes on Diophantus, Current Science 67, No. 12, December 25, 1994, pp. 957–966

    Google Scholar 

  89. W. Knorr, Problems in the interpretation of Greek number theory: Euclid and the “fundamental theorem of arithmetic”, Studies in Hist. and Philos. Sci. 7 (1976), 353–368

    Google Scholar 

  90. K. Koutsky, Eine Bemerkung zum quadratischen Charakter der Zahlen, (Czech.), Casopis 58 (1929), 42–52; FdM 55-II (1929), 691

    Google Scholar 

  91. K. Koutsky, Uber den quadratischen Charakter der Zahlen und über die Verallgemeinerung eines Satzes von Lagrange über die Verteilung der quadratischen Reste,(Czech.) Rozpravy 39 (1930), 21 pp; FdM 56-I (1930), 158

    Google Scholar 

  92. K. Koutsky, Ober die Verteilung der Potenzreste für einen Primzahlmodul, Casopis 59 (1930), 65–82 (Czech.); FdM 56-I (1930), 158

    Google Scholar 

  93. K. Koutsky, Uber die Verteilung der Potenzreste für einen Primzahlmodul. II, C. R. Congr. Math. Pays Slaves, 119–128; FdM 56-I (1930), 158

    Google Scholar 

  94. H. Kühne, Eine Wechselbeziehung zwischen Funktionen mehrerer Unbestimmter, die zu Reziprozitätsgesetzen führt, J. Reine Angew. Math. 124 (1902), 121–133; FdM 32 (1901), 210

    Google Scholar 

  95. E. Kummer, Letters to Kronecker, Collected Papers vol I, Springer 1975

    Google Scholar 

  96. J. L. Lagrange, Problèmes indéterminés du second degré, Mém. de l’Acad. Royale Berlin 23 (1769); OEuvres II, 377–535

    Google Scholar 

  97. J. L. Lagrange, Démonstration d’un théorème d’arithmétique, OEuvres de Lagrange 3 (1896), 189–201

    Google Scholar 

  98. J.H. Lambert, Aductata quaedam de numeris eorumque anatomia, Nova Acta Erudit., Leipzig 1769, 127–128

    Google Scholar 

  99. L.C. Larson, A theorem about primes proved on a chessboard, Math. Mag. 50 (1977), no. 2, 69–74

    Google Scholar 

  100. V. A. Lebesgue, Extension d’une formule de Gauss, C. R. Acad. Sci. Paris 59 (1864), 1067–1068

    Google Scholar 

  101. D.H. Lehmer, Computer technology applied to the theory of numbers, Studies Number Theory (LeVeque, ed.), 117–151 (1969)

    Google Scholar 

  102. D.H. Lehmer, E. Lehmer, W.H. Mills, Pairs of consecutive power residues, Can. J. Math. 15 (1963), 172–177

    MathSciNet  MATH  Google Scholar 

  103. E. Lehmer, Patterns of power residues, J. Number Theory 17 (1983), 37–46

    Google Scholar 

  104. H.W. Lenstra, Computing Jacobi symbols in algebraic number fields, Nieuw Arch. Wiskd. 13 (1995), 421–426

    Google Scholar 

  105. R. Linden, Uber die Verteilung quadratischer Reste, Hausarbeit Göttingen, 1976

    Google Scholar 

  106. A. Loewy, Uber die Reduktion algebraischer Gleichungen durch Adjunktion insbesondere reeller Radikale,Math. Z. 15 (1922)

    Google Scholar 

  107. D. Mahnke, Leibniz auf der Suche nach einer allgemeinen Primzahlgleichung,Bibl. Math. (3) 13 (1912/13), 26–61

    Google Scholar 

  108. M.S. Mahoney, The mathematical career of Pierre de Fermat, 16011665, 2nd ed. Princeton UP 1994

    Google Scholar 

  109. Y.I. Manin, Good proofs are proofs that make us wiser, DMV-Mitteilungen 2 (1998), 40–44

    Google Scholar 

  110. W. Mantel, Résidus quadratiques de polynômes, Nieuw Arch. Wisk. (2) 6 (1905), 374–386; FdM 36 (1905), 270

    Google Scholar 

  111. E. McClintock, On the nature and use of the functions employed on the recognition of quadratic residues, Trans. Amer. Math. Soc. 3 (1908), 92–109; FdM 33 (1902), 204;

    Google Scholar 

  112. K.D. Merrill, L.H. Walling, On quadratic reciprocity over function fields, Pacific J. Math. 173 (1996), 147–150

    MathSciNet  MATH  Google Scholar 

  113. S.M. Meyer, J.P. Sorenson, Efficient algorithms for computing Jacobi symbols, ANTS II, Proc. Bordeaux 1996, 229–243

    Google Scholar 

  114. W.H. Mills, Bounded consecutive residues and related problems, Proc. Sympos. Pure Math. 8 (1965), 170–174

    Google Scholar 

  115. F. Minding, Anfangsgründe der höheren Arithmetik, Berlin 1832

    Google Scholar 

  116. H. Minkowski, Letter to Hilbert, August 3, 1893; in: Briefe an David Hilbert, mit Beiträgen und herausgegeben von L. Rüdenberg und H. Zassenhaus, Springer, 1973

    Google Scholar 

  117. D.S. Mitrinovic, J. Sandor, B. Crstici, Handbook of number theory, Kluwer 1995

    Google Scholar 

  118. R. Mollin, A survey of Jacobi symbols, ideals, and continued fractions, Far East J. Math. Sci. 6 (1998), 355–368

    Google Scholar 

  119. M.G. Monzingo, On the distribution of quadratic residues, The Fibonacci sequence, Collect. Manuscr. (1980), 94–97

    Google Scholar 

  120. L. J. Mordell, The condition for integer solutions of ax 2 + by 2 + cz 2 + dt 2 = 0, J. Reine Angew. Math. 164 (1931), 40–49

    Google Scholar 

  121. L. J. Mordell, On the magnitude of the integer solutions of the equation ax2 + by2 + cz2 = 0, J. Number Theory 1 (1968), 1–3

    Google Scholar 

  122. Moreau, Extrait d’une lettre de M. Moreau,Nouv. Annal. (2) 12 (1873), 323–324

    Google Scholar 

  123. P. Naur, Proof versus formalization, BIT 34 (1994), 148–164

    Google Scholar 

  124. O. Neumann, Ober die Anstöße zu Kummers Schöpfung der “Idealen Complexen Zahlen”, Math. Persp. K.-R. Biermann 60th Birthday, 179–199 (1981)

    Google Scholar 

  125. O. Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc. 36 (1934), 243–274

    Google Scholar 

  126. T. Pépin, Sur certains nombres complexes compris dans la formule a + bue, J. Math. Pures Appl. (3) I (1875), 317–372

    Google Scholar 

  127. B. M. Phong, J. P. Jones, Quadratic residues and related problems, Ann. Univ. Sci. Budapest Sect. Comput. 13 (1992) 149–155

    Google Scholar 

  128. Pill A. Piltz, Ober die Häufigkeit der Primzahlen in arithmetischen Progressionen und liber verwandte Gesetze, Habilitationsschrift Jena 1884

    Google Scholar 

  129. H.C. Pocklington, The direct solution of the quadratic and cubic binomial congruences with prime moduli, Proc. Cambridge Phil. Soc. 19 (1917), 57–59

    Google Scholar 

  130. H.C. Pocklington, Quadratic and higher reciprocity of modular polynomials, Proc. Cambridge Phil. Soc. 40 (1944), 212–214

    Google Scholar 

  131. A.J. van der Poorten, P.G. Walsh, A Note on Jacobi Symbols and Continued Fractions, Amer. Math. Monthly 106 (1999), 52–56

    Article  MathSciNet  MATH  Google Scholar 

  132. C.P. Popovici, A relation between the integer part and the Euler indicatrix (Romanian), An. Univ. Bucuresti Ser. Sti. Natur. Mat.-Mec. 16 (1967), 49–52

    Google Scholar 

  133. R. Rashed, Les travaux perdus de Diophante. II, Revue Histoire Sci. 28 (1975), 3–30

    Google Scholar 

  134. L. Rédei, Über die Anzahl der Potenzreste mod p im Intervall 1, j, Nieuw Arch. Wiskunde (2) 23 (1950), 150–162

    Google Scholar 

  135. K. Reich, Zur Theorie der quadratischen Reste, Archiv Math. Phys. (II) 11 (1892), 176–192; FdM 24 (1892), 176–177

    Google Scholar 

  136. G.J. Rieger, Die Zahlentheorie bei C.F. Gauß, Gauß Gedenkband (ed.: H. Reichardt ), Teubner 1957

    Google Scholar 

  137. E. Rocci, Sulla distribuzione dei residui quadratici di un numero primo nella serie naturale, Giornale di Mat. 65 (1927), 112–134; FdM 53 (1927), 124–125

    Google Scholar 

  138. J. Rosenberg, Algebraic K-theory and its application, GTM 14’T, Springer 1996

    Google Scholar 

  139. D. Rusin, Solution of Legendre’s equation,in preparation

    Google Scholar 

  140. W. J. Ryan, Proof of the quadratic reciprocity law in primitive recursive arithmetic, Math. Scand. 45 (1979), 177–197

    Google Scholar 

  141. W. Scharlau, Richard Dedekinds algebraische Arbeiten aus seiner Göttinger Privatdozenten-Zeit 1854–1858, Abh. Braunschw. Wiss. Ges. 33 (1982), 69–70

    Google Scholar 

  142. W. Scharlau, Unveröffentlichte algebraische Arbeiten Richard Dedekinds aus seiner Göttinger Zeit 1855–1858, Arch. Hist. Exact Sci. 27 (1982), 335–367

    Google Scholar 

  143. W. Scharlau, H. Opolka, Von Fermat bis Minkowski. Eine Vorlesung fiber Zahlentheorie und ihre Entwicklung, Springer 1980; Engl. translation: From Fermat to Minkowski. Lectures on the theory of numbers and its historical development, Springer 1985

    Google Scholar 

  144. H. Scheffler, Beleuchtung und Beweis eines Satzes aus Legendre’s Zahlentheorie,Leipzig 1893?

    Google Scholar 

  145. A. Schinzel, On two conjectures of P. Chowla and S. Chowla concerning continued fractions, Ann. Mat. pura appl. 98 (1974), 111–117

    Google Scholar 

  146. F.K. Schmidt, Allgemeine Körper im Gebiet der höheren Kongruenzen, Diss. Freiburg 1925

    Google Scholar 

  147. I. Schönheim, Formules pour résoudre la congruence x 2 - a (mod P) dans des cas encore inconnus et leur application pour déterminer directement des racines primitives de certains nombres premiers (Romanian. Russian and French summary), Acad. Republ. Popul. Romine, Fil. Cluj, Studii Cerc. Mat. Fiz. 7 (1956), 51–58

    Google Scholar 

  148. R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comput. 44 (1985), 483–494

    Google Scholar 

  149. S. Schwarz, A contribution to the arithmetic of finite fields (Slovak.) Prirodoved. Priloha Techn. Obzoru Sloven. 1 no. 8 (1940), 75–81

    Google Scholar 

  150. J. Sesiano, Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusta ibn Luqa, Springer-Verlag 1982

    Google Scholar 

  151. J. Shallit, On the worst case of three algorithms for computing the Jacobi symbol, J. Symb. Comp. 10 (1990), 593–610

    Google Scholar 

  152. J. Shallit, J. Sorenson, A binary algorithm for the Jacobi symbol, SIGSAM Bull. 27 (1993), 4–11

    Article  Google Scholar 

  153. D. Shanks, Five number-theoretic algorithms, Proc. 2nd Manit. Conf. numer. Math., Winnipeg 1972, 51–70 (1973)

    Google Scholar 

  154. H. Shapiro, Introduction to the theory of numbers, Wiley and Sons 1983 [Sie] C.L. Siegel, Analytische Zahlentheorie I, Lecture Notes, Göttingen 1963

    Google Scholar 

  155. S. Singh, Bound for the solutions of a Diophantine equation in prime Galois fields, Indian J. Pure Appl. Math. 8 (1977), 1428–1430

    Google Scholar 

  156. Th. Skolem, A simple proof of the condition of solvability of the diophantine equation ax e + by 2 + cz 2 = 0, Norske Vid. Selsk. Forhdl. 24 (1952), 102–107

    Google Scholar 

  157. S.T. Smith, Quadratic residues and x 3 +y3 = z3 in models of IE l and IE 2, Notre Dame J. Formal Logic 34 (1993), 420–438

    Google Scholar 

  158. G. Speckmann, Ober die Auflösung der Congruenz x 2 - a (mod p), Arch. Math. Phys. 14 (1895), 445–448; ibid. 15 (1896), 335–336

    Google Scholar 

  159. B. Stankowitsch, Zur Theorie der Congruenzen mit einer Veränderlichen in Bezug auf einen Primzahlmodulus, Mosk. Math. Samml. X (1882); FdM 14 (1882), 124–125

    Google Scholar 

  160. S.A. Stepanov, Arithmetic of Algebraic Curves, Monographs in Contemporary Mathematics, 1994

    Google Scholar 

  161. R. V. Sterneck, Ueber die Verteilung quadratischer Reste und Nichtreste einer Primzahl, Moscow Math. Samml. 20 (1898), 269–284; FdM 29 (1898), 153–154; RSPM 7 (1898), 142

    Google Scholar 

  162. R. V. Sterneck, Sitzungsber. Akad. Wiss. Wien 2A 114 (1905), 711717

    Google Scholar 

  163. J. Stillwell, Numbers and Geometry, Undergraduate Texts in Mathematics, Springer Verlag 1997

    MATH  Google Scholar 

  164. Y. Tamarkin, A. Friedmann, Sur les congruences du second degré et les nombres de Bernoulli,Math. Ann. 62 (1906), 409–412 42 1. The Genesis of Quadratic Reciprocity

    Google Scholar 

  165. A. Tonelli, Bemerkung fiber die Auflösung quadratischer Congruenzen, Gött. Nachr. 1891, 344–346

    Google Scholar 

  166. G. Torelli, Sulla distribuzione dei resti quadratici di un numero primo, Rend. Accad. Napoli 15 (1909), 223–230; RSPM 19 (1911), 91

    Google Scholar 

  167. R. Vaidyanathaswamy, The quadratic reciprocity of polynomials modulo p, J. Indian Math. Soc. 17 (1928), 185–196

    Google Scholar 

  168. H.S. Vandiver, An aspect of the linear congruence with applications to the theory of Fermat’s quotient, Bull. Amer. Math. Soc. 22 (1915), 61–67;

    Article  MathSciNet  MATH  Google Scholar 

  169. H.S. Vandiver, On the distribution of quadratic and higher residues, Bull. Amer. Math. Soc. 31, 346–350; FdM 51 (1925), 129

    Google Scholar 

  170. H.S. Vandiver, Algorithms for the solution of the quadratic congruence, Amer. Math. Monthly 36 (1929), 83–86; FdM 55 I (1929), 95

    Google Scholar 

  171. J. Vuillemin, La Philosophie de l’algèbre, Presses Univ. France, 1962

    Google Scholar 

  172. S. Wagon, The Euclidean algorithm strikes again, Am. Math. Mon. 97 (1990), 125–129

    Google Scholar 

  173. H. Weber, Lehrbuch der Algebra 3,Braunschweig, F. Vieweg und Sohn 1908; see also http://moa.cit.cornell.edu/MOA/MOA-JOURNALS2/WEBR.html

    Google Scholar 

  174. A. Weil, L’OEuvre arithmétique d’Euler, Gedenkband L. Euler, Birkhäuser 1983

    Google Scholar 

  175. A. Weil, Number Theory, An approach through history. From Hammurapi to Legendre, Birkhäuser. XXI, 375 p. (1984)

    Google Scholar 

  176. A. L. Whiteman, On a theorem of higher reciprocity, Bull. Amer. Math. Soc. 43 (1937), 567–572

    Google Scholar 

  177. C.T. Whyburn, The density of power residues and non-residues in subintervals of [1, ~],Acta Arith 14 (1967/1968), 113–116

    Google Scholar 

  178. C.T. Whyburn, The distribution of rth powers in a finite field, J. Reine Angew. Math. 245 (1970), 183–187

    Google Scholar 

  179. K.S. Williams, On the size of a solution of Legendre’s equation, Utilitas Math. 34 (1988), 65–72

    Google Scholar 

  180. Z. Xu, Another proof of Fermat’s Last Theorem for cubes, J. Southwest Teach. Univ., Ser. B (1987), No.1, 20–22; Zbl 743. 11017

    Google Scholar 

  181. D. Zagier, A one-sentence proof that every prime p - 1 (mod 4) is a sum of two squares, Am. Math. Mon. 97 (1990), 144

    Article  MathSciNet  MATH  Google Scholar 

  182. I. Zignago, Intorno ad un teorema di aritmetica, Annali di Mat. (2) 21 (1893), 47–55

    Google Scholar 

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Lemmermeyer, F. (2000). The Genesis of Quadratic Reciprocity. In: Reciprocity Laws. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12893-0_1

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