Number theory (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Number theory" in English language version.

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ams.org

Mumford 2010, p. 387. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.
Mumford 2010, p. 388. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.

mathscinet.ams.org

The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example, Serre 1996. In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." (Hardy & Wright 2008) Serre, Jean-Pierre (1996) [1973]. A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer. ISBN 978-0-387-90040-7. Hardy, Godfrey Harold; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243.

archive.org

The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example, Serre 1996. In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." (Hardy & Wright 2008) Serre, Jean-Pierre (1996) [1973]. A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer. ISBN 978-0-387-90040-7. Hardy, Godfrey Harold; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243.
Boyer & Merzbach 1991, p. 82. Boyer, Carl Benjamin; Merzbach, Uta C. (1991) [1968]. A History of Mathematics (2nd ed.). New York: Wiley. ISBN 978-0-471-54397-8. 1968 edition at archive.org
Heath 1921, p. 76. Heath, Thomas L. (1921). A History of Greek Mathematics, Volume 1: From Thales to Euclid. Oxford: Clarendon Press. Retrieved 2016-02-28.
Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: Plofker 2008, pp. 134–140. See also Clark 1930, pp. 42–50. A slightly more explicit description of the kuṭṭaka was later given in Brahmagupta, Brāhmasphuṭasiddhānta, XVIII, 3–5 (in Colebrooke 1817, p. 325, cited in Clark 1930, p. 42). Plofker, Kim (2008). Mathematics in India. Princeton University Press. ISBN 978-0-691-12067-6. Aryabhata (1930). The Āryabhaṭīya of Āryabhaṭa: An ancient Indian work on Mathematics and Astronomy. Translated by Clark, Walter Eugene. University of Chicago Press. Retrieved 2016-02-28. Colebrooke, Henry Thomas (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. London: J. Murray. Retrieved 2016-02-28. Aryabhata (1930). The Āryabhaṭīya of Āryabhaṭa: An ancient Indian work on Mathematics and Astronomy. Translated by Clark, Walter Eugene. University of Chicago Press. Retrieved 2016-02-28.
Colebrooke 1817, p. lxv, cited in Hopkins 1990, p. 302. See also the preface in Sachau & Bīrūni 1888 cited in Smith 1958, pp. 168 Colebrooke, Henry Thomas (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. London: J. Murray. Retrieved 2016-02-28. Hopkins, J. F. P. (1990). "Geographical and Navigational Literature". In Young, M. J. L.; Latham, J. D.; Serjeant, R. B. (eds.). Religion, Learning and Science in the 'Abbasid Period. The Cambridge history of Arabic literature. Cambridge University Press. ISBN 978-0-521-32763-3. Sachau, Eduard; Bīrūni, ̄Muḥammad ibn Aḥmad (1888). Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1. London: Kegan, Paul, Trench, Trübner & Co. Archived from the original on 2016-03-03. Retrieved 2016-02-28. Smith, D. E. (1958). History of Mathematics, Vol I. New York: Dover.

arxiv.org

Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis". Reviews of Modern Physics. 83 (2): 307–330. arXiv:1101.3116. Bibcode:2011RvMP...83..307S. doi:10.1103/RevModPhys.83.307. S2CID 119290777.

books.google.com

See, for example, Sunzi Suanjing, Ch. 3, Problem 36, in Lam & Ang 2004, pp. 223–224:

[36] Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. Answer: Male.

Method: Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female.

This is the last problem in Sunzi's otherwise matter-of-fact treatise.
Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28.
Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way (Weil 1984, pp. 159, 161). (There were already some recognisable features of professional practice, viz., seeking correspondents, visiting foreign colleagues, building private libraries (Weil 1984, pp. 160–161). Matters started to shift in the late seventeenth century (Weil 1984, p. 161); scientific academies were founded in England (the Royal Society, 1662) and France (the Académie des sciences, 1666) and Russia (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 (Weil 1984, p. 163 and Varadarajan 2006, p. 7). In this context, the term amateur usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy (Truesdell 1984, p. xv); cited in Varadarajan 2006, p. 9). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28. Truesdell, C. A. (1984). "Leonard Euler, Supreme Geometer". Leonard Euler, Elements of Algebra. Translated by Hewlett, John (reprint of 1840 5th ed.). New York: Springer-Verlag. ISBN 978-0-387-96014-2. This Google books preview of Elements of algebra lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book: Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, Iwaniec & Kowalski 2004 or Montgomery & Vaughan 2007 Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3633-0. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory: I, Classical Theory. Cambridge University Press. ISBN 978-0-521-84903-6. Retrieved 2016-02-28.
- Lozano-Robledo 2019, p. xiii
- - Wilson 2020, pp. 1–2
  - Duverney 2010, p. v
    Pomerance & Sárközy 1995, p. 969
    Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Huffman 2011. On Thales, see Eudemus ap. Proclus, 65.7, (for example, Morrow 1992, p. 52) cited in: O'Grady 2004, p. 1. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, Morrow 1992, p. xxx on Proclus's reliability. Huffman, Carl A. (8 August 2011). "Pythagoras". In Zalta, Edward N. (ed.). Stanford Encyclopaedia of Philosophy (Fall 2011 ed.). Archived from the original on 2 December 2013. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7. O'Grady, Patricia (September 2004). "Thales of Miletus". The Internet Encyclopaedia of Philosophy. Archived from the original on 6 January 2016. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7.
    Sunzi Suanjing, Chapter 3, Problem 26. This can be found in Lam & Ang 2004, pp. 219–220, which contains a full translation of the Suan Ching (based on Qian 1963). See also the discussion in Lam & Ang 2004, pp. 138–140. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28. Qian, Baocong, ed. (1963). Suanjing shi shu (Ten Mathematical Classics) (in Chinese). Beijing: Zhonghua shuju. Archived from the original on 2013-11-02. Retrieved 2016-02-28. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28.
    The date of the text has been narrowed down to 220–420 AD (Yan Dunjie) or 280–473 AD (Wang Ling) through internal evidence (= taxation systems assumed in the text). See Lam & Ang 2004, pp. 27–28. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28.
    Weil 1984, pp. 45–46. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Faulkner, Nicholas; Hosch, William L. (2017). "Numbers and Measurements". Encyclopaedia Britannica. ISBN 978-1-5383-0042-8. Retrieved 2019-08-06.
    Weil 1984, p. 92. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Weil 1984, pp. 2, 172. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Varadarajan 2006, p. 9. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
    Weil 1984, pp. 1–2. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Weil 1984, p. 2 and Varadarajan 2006, p. 37 Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
    Weil 1984, pp. 178–179. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Varadarajan 2006, p. 39 and Weil 1984, pp. 176–189 Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Weil 1984, p. 174. Euler was generous in giving credit to others (Varadarajan 2006, p. 14), not always correctly. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
    Weil 1984, p. 183. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Varadarajan 2006, pp. 45–55; see also chapter III. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
    Weil 1984, pp. 327–328. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Weil 1984, pp. 337–338. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Weil 1984, pp. 332–334. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Goldstein & Schappacher 2007, p. 14. Goldstein, Catherine; Schappacher, Norbert (2007). "A book in search of a discipline". In Goldstein, C.; Schappacher, N.; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's "Disquisitiones Arithmeticae". Berlin & Heidelberg: Springer. pp. 3–66. ISBN 978-3-540-20441-1. Retrieved 2016-02-28.
    From the preface of Disquisitiones Arithmeticae; the translation is taken from Goldstein & Schappacher 2007, p. 16 Goldstein, Catherine; Schappacher, Norbert (2007). "A book in search of a discipline". In Goldstein, C.; Schappacher, N.; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's "Disquisitiones Arithmeticae". Berlin & Heidelberg: Springer. pp. 3–66. ISBN 978-3-540-20441-1. Retrieved 2016-02-28.
    See the discussion in section 5 of Goldstein & Schappacher 2007. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in Weil 1984, p. 25). Goldstein, Catherine; Schappacher, Norbert (2007). "A book in search of a discipline". In Goldstein, C.; Schappacher, N.; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's "Disquisitiones Arithmeticae". Berlin & Heidelberg: Springer. pp. 3–66. ISBN 978-3-540-20441-1. Retrieved 2016-02-28. Weil, André (1984). Number Theory: an Approach Through History – from Hammurapi to Legendre. Boston: Birkhäuser. ISBN 978-0-8176-3141-3. Retrieved 2016-02-28.
    Apostol 1976, p. 7. Apostol, Tom M. (1976). Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer. ISBN 978-0-387-90163-3. Retrieved 2016-02-28.
    Varadarajan 2006, sections 2.5, 3.1 and 6.1. Varadarajan, V. S. (2006). Euler Through Time: A New Look at Old Themes. American Mathematical Society. ISBN 978-0-8218-3580-7. Retrieved 2016-02-28.
    Granville 2008, pp. 322–348. Granville, Andrew (2008). "Analytic number theory". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. ISBN 978-0-691-11880-2. Retrieved 2016-02-28.
    Richmond & Richmond (2009), Section 3.4 (Divisibility Tests), p. 102–108
    Granville 2008, section 1: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for good approximations." Granville, Andrew (2008). "Analytic number theory". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. ISBN 978-0-691-11880-2. Retrieved 2016-02-28.
    Granville 2008, section 3: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]" Granville, Andrew (2008). "Analytic number theory". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. ISBN 978-0-691-11880-2. Retrieved 2016-02-28.
    See, for example, Montgomery & Vaughan 2007, p. 1. Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative Number Theory: I, Classical Theory. Cambridge University Press. ISBN 978-0-521-84903-6. Retrieved 2016-02-28.
    Edwards 2000, p. 79. Edwards, Harold M. (2000) [1977]. Fermat's Last Theorem: a Genetic Introduction to Algebraic Number Theory. Graduate Texts in Mathematics. Vol. 50 (reprint of 1977 ed.). Springer Verlag. ISBN 978-0-387-95002-0.
    Bryant, John; Sangwin, Christopher J. (2008). How Round is Your Circle?: Where Engineering and Mathematics Meet. Princeton University Press. p. 178. ISBN 978-0-691-13118-4.
    Hardy, Godfrey Harold (2012) [1940]. A Mathematician's Apology. Cambridge University Press. p. 140. ISBN 978-0-521-42706-7. OCLC 922010634. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.
    Kraft, James S.; Washington, Lawrence C. (2014). Elementary Number Theory. Textbooks in mathematics. CRC Press. p. 7. ISBN 978-1-4987-0269-0.
    britannica.com
    "Number theory | Definition, Topics, & History | Britannica". www.britannica.com. Retrieved 2025-06-28.
    cam.ac.uk
    hps.cam.ac.uk
    Robson 2001, p. 201. This is controversial. See Plimpton 322. Robson's article is written polemically (Robson 2001, p. 202) with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" (Robson 2001, p. 167); at the same time, it settles to the conclusion that
    [...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (Robson 2001, p. 202).
    
    Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".(Robson 2001, pp. 199–200)
    Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Neugebauer & Sachs 1945, p. 40. The term takiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Robson 2001, p. 192 Neugebauer, Otto E.; Sachs, Abraham Joseph; Götze, Albrecht (1945). Mathematical Cuneiform Texts. American Oriental Series. Vol. 29. American Oriental Society etc. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Robson 2001, p. 189. Other sources give the modern formula $(p^{2}-q^{2},2pq,p^{2}+q^{2})$ . Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.(van der Waerden 1961, p. 79) Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. van der Waerden, Bartel L. (1961). Science Awakening. Vol. 1 or 2. Translated by Dresden, Arnold. New York: Oxford University Press.
    columbia.edu
    math.columbia.edu
    Goldfeld 2003. Goldfeld, Dorian M. (2003). "Elementary Proof of the Prime Number Theorem: a Historical Perspective" (PDF). Archived (PDF) from the original on 2016-03-03. Retrieved 2016-02-28.
    doi.org
    doi.org
    Robson 2001, p. 201. This is controversial. See Plimpton 322. Robson's article is written polemically (Robson 2001, p. 202) with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" (Robson 2001, p. 167); at the same time, it settles to the conclusion that
    [...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (Robson 2001, p. 202).
    
    Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".(Robson 2001, pp. 199–200)
    Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Neugebauer & Sachs 1945, p. 40. The term takiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Robson 2001, p. 192 Neugebauer, Otto E.; Sachs, Abraham Joseph; Götze, Albrecht (1945). Mathematical Cuneiform Texts. American Oriental Series. Vol. 29. American Oriental Society etc. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Robson 2001, p. 189. Other sources give the modern formula $(p^{2}-q^{2},2pq,p^{2}+q^{2})$ . Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.(van der Waerden 1961, p. 79) Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. van der Waerden, Bartel L. (1961). Science Awakening. Vol. 1 or 2. Translated by Dresden, Arnold. New York: Oxford University Press.
    Friberg 1981, p. 302. Friberg, Jöran (August 1981). "Methods and Traditions of Babylonian Mathematics: Plimpton 322, Pythagorean Triples and the Babylonian Triangle Parameter Equations". Historia Mathematica. 8 (3): 277–318. doi:10.1016/0315-0860(81)90069-0.
    Rashed 1980, pp. 305–321. Rashed, Roshdi (1980). "Ibn al-Haytham et le théorème de Wilson". Archive for History of Exact Sciences. 22 (4): 305–321. doi:10.1007/BF00717654. S2CID 120885025.
    An Introduction to Number Theory with Cryptography (2nd ed.). Chapman and Hall/CRC. 2018. doi:10.1201/9781351664110. ISBN 978-1-351-66411-0. Archived from the original on 2023-03-01. Retrieved 2023-02-22.
    Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis". Reviews of Modern Physics. 83 (2): 307–330. arXiv:1101.3116. Bibcode:2011RvMP...83..307S. doi:10.1103/RevModPhys.83.307. S2CID 119290777.
    Baylis, John (2018). Error-Correcting Codes: A Mathematical Introduction. Routledge. doi:10.1201/9780203756676. ISBN 978-0-203-75667-6. Retrieved 2023-02-22.
    Livné, R. (2001), Ciliberto, Ciro; Hirzebruch, Friedrich; Miranda, Rick; Teicher, Mina (eds.), "Communication Networks and Hilbert Modular Forms", Applications of Algebraic Geometry to Coding Theory, Physics and Computation, Dordrecht: Springer, pp. 255–270, doi:10.1007/978-94-010-1011-5_13, ISBN 978-1-4020-0005-8, retrieved 2023-02-22
    Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). "Aesthetics, Dynamics, and Musical Scales: A Golden Connection". Journal of New Music Research. 31 (1): 51–58. doi:10.1076/jnmr.31.1.51.8099. hdl:10261/18003. ISSN 0929-8215. S2CID 12232457.
    dx.doi.org
    Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). "Aesthetics, Dynamics, and Musical Scales: A Golden Connection". Journal of New Music Research. 31 (1): 51–58. doi:10.1076/jnmr.31.1.51.8099. hdl:10261/18003. ISSN 0929-8215. S2CID 12232457.
    encyclopediaofmath.org
    Karatsuba, A.A. (2020). "Number theory". Encyclopedia of Mathematics. Springer. Retrieved 2025-05-03.
    Bukhshtab, A.A. (2014). "Elementary number theory". Encyclopedia of Mathematics. Springer. Retrieved 2025-05-03.
    Karatsuba, A.A. (2014-10-18). "Analytic number theory". Encyclopedia of Mathematics.
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    Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). "Aesthetics, Dynamics, and Musical Scales: A Golden Connection". Journal of New Music Research. 31 (1): 51–58. doi:10.1076/jnmr.31.1.51.8099. hdl:10261/18003. ISSN 0929-8215. S2CID 12232457.
    harvard.edu
    ui.adsabs.harvard.edu
    Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis". Reviews of Modern Physics. 83 (2): 307–330. arXiv:1101.3116. Bibcode:2011RvMP...83..307S. doi:10.1103/RevModPhys.83.307. S2CID 119290777.
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    Milne 2017, p. 2. Milne, J. S. (18 March 2017). "Algebraic Number Theory". Retrieved 7 April 2020.
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    Long 1972, p. 1. Long, Calvin T. (1972). Elementary Introduction to Number Theory (2nd ed.). Lexington, VA: D.C. Heath and Company. LCCN 77171950.
    sciencedirect.com
    Page 2003, pp. 18–19, 34
    Page 2003, p. 34
    Page 2003, pp. 34–35
    scribd.com
    Sunzi Suanjing, Chapter 3, Problem 26. This can be found in Lam & Ang 2004, pp. 219–220, which contains a full translation of the Suan Ching (based on Qian 1963). See also the discussion in Lam & Ang 2004, pp. 138–140. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28. Qian, Baocong, ed. (1963). Suanjing shi shu (Ten Mathematical Classics) (in Chinese). Beijing: Zhonghua shuju. Archived from the original on 2013-11-02. Retrieved 2016-02-28. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28.
    semanticscholar.org
    api.semanticscholar.org
    Rashed 1980, pp. 305–321. Rashed, Roshdi (1980). "Ibn al-Haytham et le théorème de Wilson". Archive for History of Exact Sciences. 22 (4): 305–321. doi:10.1007/BF00717654. S2CID 120885025.
    Schumayer, Daniel; Hutchinson, David A. W. (2011). "Physics of the Riemann Hypothesis". Reviews of Modern Physics. 83 (2): 307–330. arXiv:1101.3116. Bibcode:2011RvMP...83..307S. doi:10.1103/RevModPhys.83.307. S2CID 119290777.
    Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). "Aesthetics, Dynamics, and Musical Scales: A Golden Connection". Journal of New Music Research. 31 (1): 51–58. doi:10.1076/jnmr.31.1.51.8099. hdl:10261/18003. ISSN 0929-8215. S2CID 12232457.
    springer.com
    link.springer.com
    Livné, R. (2001), Ciliberto, Ciro; Hirzebruch, Friedrich; Miranda, Rick; Teicher, Mina (eds.), "Communication Networks and Hilbert Modular Forms", Applications of Algebraic Geometry to Coding Theory, Physics and Computation, Dordrecht: Springer, pp. 255–270, doi:10.1007/978-94-010-1011-5_13, ISBN 978-1-4020-0005-8, retrieved 2023-02-22
    stanford.edu
    plato.stanford.edu
    Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Huffman 2011. On Thales, see Eudemus ap. Proclus, 65.7, (for example, Morrow 1992, p. 52) cited in: O'Grady 2004, p. 1. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, Morrow 1992, p. xxx on Proclus's reliability. Huffman, Carl A. (8 August 2011). "Pythagoras". In Zalta, Edward N. (ed.). Stanford Encyclopaedia of Philosophy (Fall 2011 ed.). Archived from the original on 2 December 2013. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7. O'Grady, Patricia (September 2004). "Thales of Miletus". The Internet Encyclopaedia of Philosophy. Archived from the original on 6 January 2016. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7.
    taylorfrancis.com
    An Introduction to Number Theory with Cryptography (2nd ed.). Chapman and Hall/CRC. 2018. doi:10.1201/9781351664110. ISBN 978-1-351-66411-0. Archived from the original on 2023-03-01. Retrieved 2023-02-22.
    Baylis, John (2018). Error-Correcting Codes: A Mathematical Introduction. Routledge. doi:10.1201/9780203756676. ISBN 978-0-203-75667-6. Retrieved 2023-02-22.
    upenn.edu
    onlinebooks.library.upenn.edu
    Colebrooke 1817, p. lxv, cited in Hopkins 1990, p. 302. See also the preface in Sachau & Bīrūni 1888 cited in Smith 1958, pp. 168 Colebrooke, Henry Thomas (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. London: J. Murray. Retrieved 2016-02-28. Hopkins, J. F. P. (1990). "Geographical and Navigational Literature". In Young, M. J. L.; Latham, J. D.; Serjeant, R. B. (eds.). Religion, Learning and Science in the 'Abbasid Period. The Cambridge history of Arabic literature. Cambridge University Press. ISBN 978-0-521-32763-3. Sachau, Eduard; Bīrūni, ̄Muḥammad ibn Aḥmad (1888). Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1. London: Kegan, Paul, Trench, Trübner & Co. Archived from the original on 2016-03-03. Retrieved 2016-02-28. Smith, D. E. (1958). History of Mathematics, Vol I. New York: Dover.
    utm.edu
    iep.utm.edu
    Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Huffman 2011. On Thales, see Eudemus ap. Proclus, 65.7, (for example, Morrow 1992, p. 52) cited in: O'Grady 2004, p. 1. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, Morrow 1992, p. xxx on Proclus's reliability. Huffman, Carl A. (8 August 2011). "Pythagoras". In Zalta, Edward N. (ed.). Stanford Encyclopaedia of Philosophy (Fall 2011 ed.). Archived from the original on 2 December 2013. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7. O'Grady, Patricia (September 2004). "Thales of Miletus". The Internet Encyclopaedia of Philosophy. Archived from the original on 6 January 2016. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7.
    web.archive.org
    Robson 2001, p. 201. This is controversial. See Plimpton 322. Robson's article is written polemically (Robson 2001, p. 202) with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" (Robson 2001, p. 167); at the same time, it settles to the conclusion that
    [...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems (Robson 2001, p. 202).
    
    Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".(Robson 2001, pp. 199–200)
    Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Neugebauer & Sachs 1945, p. 40. The term takiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Robson 2001, p. 192 Neugebauer, Otto E.; Sachs, Abraham Joseph; Götze, Albrecht (1945). Mathematical Cuneiform Texts. American Oriental Series. Vol. 29. American Oriental Society etc. Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21.
    Robson 2001, p. 189. Other sources give the modern formula $(p^{2}-q^{2},2pq,p^{2}+q^{2})$ . Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.(van der Waerden 1961, p. 79) Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322" (PDF). Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Archived from the original (PDF) on 2014-10-21. van der Waerden, Bartel L. (1961). Science Awakening. Vol. 1 or 2. Translated by Dresden, Arnold. New York: Oxford University Press.
    Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Huffman 2011. On Thales, see Eudemus ap. Proclus, 65.7, (for example, Morrow 1992, p. 52) cited in: O'Grady 2004, p. 1. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, Morrow 1992, p. xxx on Proclus's reliability. Huffman, Carl A. (8 August 2011). "Pythagoras". In Zalta, Edward N. (ed.). Stanford Encyclopaedia of Philosophy (Fall 2011 ed.). Archived from the original on 2 December 2013. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7. O'Grady, Patricia (September 2004). "Thales of Miletus". The Internet Encyclopaedia of Philosophy. Archived from the original on 6 January 2016. Retrieved 7 February 2012. Euclid; Proclus (1992). A Commentary on Book 1 of Euclid's Elements. Translated by Morrow, Glenn Raymond. Princeton University Press. ISBN 978-0-691-02090-7.
    Sunzi Suanjing, Chapter 3, Problem 26. This can be found in Lam & Ang 2004, pp. 219–220, which contains a full translation of the Suan Ching (based on Qian 1963). See also the discussion in Lam & Ang 2004, pp. 138–140. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28. Qian, Baocong, ed. (1963). Suanjing shi shu (Ten Mathematical Classics) (in Chinese). Beijing: Zhonghua shuju. Archived from the original on 2013-11-02. Retrieved 2016-02-28. Lam, Lay Yong; Ang, Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised ed.). Singapore: World Scientific. ISBN 978-981-238-696-0. Retrieved 2016-02-28.
    Mumford 2010, p. 387. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.
    Mumford 2010, p. 388. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.
    Colebrooke 1817, p. lxv, cited in Hopkins 1990, p. 302. See also the preface in Sachau & Bīrūni 1888 cited in Smith 1958, pp. 168 Colebrooke, Henry Thomas (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. London: J. Murray. Retrieved 2016-02-28. Hopkins, J. F. P. (1990). "Geographical and Navigational Literature". In Young, M. J. L.; Latham, J. D.; Serjeant, R. B. (eds.). Religion, Learning and Science in the 'Abbasid Period. The Cambridge history of Arabic literature. Cambridge University Press. ISBN 978-0-521-32763-3. Sachau, Eduard; Bīrūni, ̄Muḥammad ibn Aḥmad (1888). Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1. London: Kegan, Paul, Trench, Trübner & Co. Archived from the original on 2016-03-03. Retrieved 2016-02-28. Smith, D. E. (1958). History of Mathematics, Vol I. New York: Dover.
    Goldfeld 2003. Goldfeld, Dorian M. (2003). "Elementary Proof of the Prime Number Theorem: a Historical Perspective" (PDF). Archived (PDF) from the original on 2016-03-03. Retrieved 2016-02-28.
    An Introduction to Number Theory with Cryptography (2nd ed.). Chapman and Hall/CRC. 2018. doi:10.1201/9781351664110. ISBN 978-1-351-66411-0. Archived from the original on 2023-03-01. Retrieved 2023-02-22.
    worldcat.org
    search.worldcat.org
    Mumford 2010, p. 387. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.
    Mumford 2010, p. 388. Mumford, David (March 2010). "Mathematics in India: reviewed by David Mumford" (PDF). Notices of the American Mathematical Society. 57 (3): 387. ISSN 1088-9477. Archived (PDF) from the original on 2021-05-06. Retrieved 2021-04-28.
    Hardy, Godfrey Harold (2012) [1940]. A Mathematician's Apology. Cambridge University Press. p. 140. ISBN 978-0-521-42706-7. OCLC 922010634. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.
    Cartwright, Julyan H. E.; Gonzalez, Diego L.; Piro, Oreste; Stanzial, Domenico (2002-03-01). "Aesthetics, Dynamics, and Musical Scales: A Golden Connection". Journal of New Music Research. 31 (1): 51–58. doi:10.1076/jnmr.31.1.51.8099. hdl:10261/18003. ISSN 0929-8215. S2CID 12232457.
    
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