Number theory (or arithmetic or higher arithmetic in older usage)

8 min readMay 29, 2021

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, “Mathematics is the queen of the sciences — and number theory is the queen of mathematics.”[1][note 1] Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by “number theory”.[note 2] (The word “arithmetic” is used by the general public to mean “elementary calculations”; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence.[note 3] In particular, arithmetical is commonly preferred as an adjective to number-theoretic.

Contents
1 History
1.1 Origins
1.1.1 Dawn of arithmetic
1.1.2 Classical Greece and the early Hellenistic period
1.1.3 Diophantus
1.1.4 Āryabhaṭa, Brahmagupta, Bhāskara
1.1.5 Arithmetic in the Islamic golden age
1.1.6 Western Europe in the Middle Ages
1.2 Early modern number theory
1.2.1 Fermat
1.2.2 Euler
1.2.3 Lagrange, Legendre, and Gauss
1.3 Maturity and division into subfields
2 Main subdivisions
2.1 Elementary number theory
2.2 Analytic number theory
2.3 Algebraic number theory
2.4 Diophantine geometry
3 Other subfields
3.1 Probabilistic number theory
3.2 Arithmetic combinatorics
3.3 Computational number theory
4 Applications
5 Prizes
6 See also
7 Notes
8 References
9 Sources
10 Further reading
11 External links
History
Origins
Dawn of arithmetic

The Plimpton 322 tablet
The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BCE) contains a list of “Pythagorean triples”, that is, integers {\displaystyle (a,b,c)}(a,b,c) such that {\displaystyle a^{2}+b^{2}=c^{2}}a^{2}+b^{2}=c^{2}. The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: “The takiltum of the diagonal which has been subtracted such that the width…”[2]

The table’s layout suggests[3] that it was constructed by means of what amounts, in modern language, to the identity

{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}{\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}
which is implicit in routine Old Babylonian exercises.[4] If some other method was used,[5] the triples were first constructed and then reordered by {\displaystyle c/a}c/a, presumably for actual use as a “table”, for example, with a view to applications.

It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems.[6][note 4]

While Babylonian number theory — or what survives of Babylonian mathematics that can be called thus — consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of “algebra”) was exceptionally well developed.[7] Late Neoplatonic sources[8] state that Pythagoras learned mathematics from the Babylonians. Much earlier sources[9] state that Thales and Pythagoras traveled and studied in Egypt.

Euclid IX 21–34 is very probably Pythagorean;[10] it is very simple material (“odd times even is even”, “if an odd number measures [= divides] an even number, then it also measures [= divides] half of it”), but it is all that is needed to prove that {\displaystyle {\sqrt {2}}}{\sqrt {2}} is irrational.[11] Pythagorean mystics gave great importance to the odd and the even.[12] The discovery that {\displaystyle {\sqrt {2}}}{\sqrt {2}} is irrational is credited to the early Pythagoreans (pre-Theodorus).[13] By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.[14] This forced a distinction between numbers (integers and the rationals — the subjects of arithmetic), on the one hand, and lengths and proportions (which we would identify with real numbers, whether rational or not), on the other hand.

The Pythagorean tradition spoke also of so-called polygonal or figurate numbers.[15] While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th century).

We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both. The Chinese remainder theorem appears as an exercise [16] in Sunzi Suanjing (3rd, 4th or 5th century CE.)[17] (There is one important step glossed over in Sunzi’s solution:[note 5] it is the problem that was later solved by Āryabhaṭa’s Kuṭṭaka — see below.)

There is also some numerical mysticism in Chinese mathematics,[note 6] but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans’ perfect numbers, magic squares have passed from superstition into recreation.

Classical Greece and the early Hellenistic period
Further information: Ancient Greek mathematics
Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period.[18] In the case of number theory, this means, by and large, Plato and Euclid, respectively.

While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition.

Eusebius, PE X, chapter 4 mentions of Pythagoras:

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Number theory (or arithmetic or higher arithmetic in older usage)

Written by Mafor