User:Virginia-American/Sandbox/References

See also

• Euler's criterion
• Gauss's lemma
• Zolotarev's lemma
• Law of quadratic reciprocity
• Character sum
• Paley graph

References

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

• Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0387962549 {{citation}}: |first2= has generic name (help)

• Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 {{citation}}: |first2= has generic name (help)

The two monographs Gauss published on biquadratic reciprocity have consecutively-numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".

• Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6

• Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7

These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148

German translations are in pp. 511–533 and 534–586 of the following, which also has the Disquisitiones Arithmeticae and Gauss's other papers on number theory.

• Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 {{citation}}: |first2= has generic name (help)

• Bach, Eric; Shallit, Jeffrey (1966), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02045-5 {{citation}}: Check |isbn= value: checksum (help)

• Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: Springer, ISBN 3-540-55640-0

• Cox, David A. (1989), Primes of the form x² + n y², New York: Wiley, ISBN 0-471-50654-0

• Crandall, Richard; Pomeramce, Carl (2001), Prime Numbers: A Computatinal Perspective, New York: Springer, ISBN 0-387-04777-9 {{citation}}: Check |isbn= value: checksum (help)

• Davenport, Harold (2000). Multiplicative Number Theory (Third edition). New York: Springer. ISBN 0-387-95097-4.

• Edwards, Harold (1977). Fermat's Last Theorem. New York: Springer. ISBN 0-387-90230-9.

• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A7.1: AN1, pg.249.}}

• Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma: Addison-Wesley, ISBN 0-201-55802-5

• Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and work, Providence RI: AMS / Chelsea, ISBN 978-0821820230

• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0198531715

• Hatcher, Allen (2022), Topology of Numbers, Providence, RI: American Mathematical Society, ISBN 978-1470456115

• Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X

• Khinchin, A. Ya. (1998), Three Pearls of Number Theory, Mineola, NY: Dover, ISBN 978-0486400266

• Knopfmacher, John (1990), Abstract Analytic Number Theory, New York: Dover, ISBN 0-486-66344-2

• Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea

• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66967-4 {{citation}}: Check |isbn= value: checksum (help)

• Manders, Kenneth L. (1978). "NP-Complete Decision Problems for Binary Quadratics". Journal of Computer and System Sciences. 16 (2): 168–184. doi:10.1016/0022-0000(78)90044-2. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Nathanson, Melvyn B. (1996), Additive Number Theory The Classical Bases, Berlin: Springer, ISBN 978-0387946566

• Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0821820766

• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5

• Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5

• Sandifer, Charles (2007), The early mathematics of Leonhard Euler, MAA, ISBN 0-883-85559-3

• Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986), The Theory of the Riemann Zeta-function (2nd ed.), Oxford: Oxford U. P., ISBN 0-19-853369-1

• Williams, Kenneth S. (2011), Number Theory in the Spirit of Liouville, Cambridge: Cambridge U. P., ISBN 978-0-521-17562-3

External links

• MathWorld: Quadratic Residue
• PlanetMath: Proof of Pólya-Vinogradov inequality
• The Euler Archive