Draft:History of number theory

originally forked from the history section in the number theory article

Elementary number theory: find an integer with remainders 2, 3, 2 when divided by 3, 5, 7, respectively.

Analytic number theory: do all nontrivial zeroes of the zeta function lie on the critical line?

The history of number theory covers the development of the mathematical subfield. It focusses on number theory but also discusses related topics on arithmetic and numerology. One of the oldest branches of mathematics, number theory arose out of problems related to multiplications and division of integers.

From the ancient to post-classical periods, development was done independently around the world, starting in Babylon and Egypt. The ancient Greeks formalised number theory as a study. Their study included integer classes, divisibility, prime numbers, factorisation and Diophantine equations. Chinese mathematicians studied number theory for astronomy and the calendar, with their work culminating in the Chinese remainder theorem. In India, mathematicians innovated the use of zero and negative numbers and first studied Pell's equations. Development shifted to the Islamic world in the post-classical, expanding on Greek works.

Number theory witnessed a resurgence in Europe following the contributions of Pierre de Fermat (1601-1665), albeit it was again overshadowed by the development of calculus. He famously conjectured what would become Fermat's Last Theorem, and studied prime numbers. Leonhard Euler (1707-1783) authored over one thousand pages about number theory, frequently solving Fermat's assertions and extending ancient Greek works. Three European contemporaries continued the work in elementary number theory: Joseph-Louis Lagrange proved the four-square theorem and Wilson's theorem, Adrien-Marie Legendre proved specific cases of Fermat's Last Theorem, and Carl Friedrich Gauss introduced congruences.

Euler's study of the zeta function formed a basis for the subfield of analytic number theory. Peter Gustav Lejeune Dirichlet (1805-1859) pioneered the subfield's methods with a proof of his analytic theorem on arithmetic progressions. The subfield studies the distribution of primes and seeks for a function that approximates it. For example, ${\textstyle {\frac {x}{\log x}}}$ would be proved by the end of the 19th century. Bernhard Riemann (1826-1866) found a connection between the zeta function and the prime-counting function, which he conjectured to be valid. By the 20th century, a plethora of subfields had had emerged, including algebraic, geometric, and combinatorial number theory. Algebraic number theory closed the millennium with the proof of Fermat's Last Theorem by Andrew Wiles.

Open questions remain, such as the Riemann hypothesis that is among the seven Millennium Prize Problems, Goldbach's conjecture on the representation of even numbers as the sum of two primes, and the existence of odd perfect numbers. Number theory was once regarded as the canonical example of pure mathematics with no applications outside the field. In 1970s, it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms. Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors. Elementary number theory is taught in discrete mathematics courses for computer scientists.

Conception

Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts.

The history of number theory is the study of the historical development of the mathematical subfield. Number theory studies the structure and properties of integers as well as the relations and laws between them.^[1] It arose out of problems related to multiplications and division of integers and is one of the oldest branches of mathematics alongside geometry. Its pre-modern period covers independent developments until the post-classical period. Modern number theory began in the 17th century. Influential subfields include elementary, analytic, algebraic, and geometric number theory.^[2]

Number theory is closely related to arithmetic and some authors use the terms as synonyms.^[3] In contemporary history, arithmetic is the study of numerical operations that extends to the real numbers. The ancient Greeks distinguished between "arithmētĭkḗ" and "logistikḗ". "Arithmētĭkḗ" is synonymous with the modern English "number theory", while "logistikḗ" is equivalent to "arithmetic", the practical use of numerical calculations. Number theory was reserved to scholars and royality.^[4] In a more specific sense, number theory is restricted to the theoretical study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality.^[5] Traditionally, it is known as higher arithmetic.^[6] The earliest appearance of the term number theory was in the 19th century.^[7]

Another field that evolved alongside number theory is numerology. It treats numbers as objects of mystical properties. Pythagoreanism was a school of thought in ancient Greece that explored number theory on the premise of numerology. An influential type of numerology is gematria, which assigns numbers to alphabetical letters. The sum of letters in a word was said to reveal secrets, predict the future, and mark someone as evil. Numerology was not clearly distinguished from mathematics and is considered a pseudoscience in contemporary history.^[8]

The earliest forms of arithmetic are sometimes traced back to counting and tally marks used to keep track of quantities. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.^[9] Some claim that tally marks on the Ishango bone list a sequence of prime numbers. Opponents argue that the concept of division, a precedent for prime numbers, evolved only after the rise of agriculture around 10,000 BC.^[10] However, a basic sense of numbers may predate these findings and might even have existed before the development of language.^[11]

Ancient to post-classical

A period of pre-modern developments in number theory is generally held to range from the dawn of civilisation to the 17th century. It covers fundamental concepts and theorems that rely on elementary proofs.^[12] Development was done independently in cradles of civilisation in different around the globe. Early developments took place in Babylonia and Egypt. Influential developments include Greece, China, India, and the Islamic world.^[13] Numerology motivated mathematicians from different cultures to study the nature of numbers.^[14]

Early development

The Babylonians were able to produce large Pythagorean triples.

The development of number theory occurred independently and a systematic study did not exist. The knowledge of numbers existed in the early civilisations of Mesopotamia, Egypt, China, and India. Surviving sources take the form of tablets, papyri, and carvings. The first positional numeral system was developed by the Babylonians starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient. They may have been familiar with prime factorisation.^[15]

A famous Babylonian artefact of number theory is Plimpton 322. It is a fragment of a larger clay tablet, dated around 1800 BC, that contains a list of fifteen Pythagorean triples. A Pythagorean triple are three integers ${\displaystyle x,y,z}$ that satisfy the Pythagorean equation ${\displaystyle x^{2}+y^{2}=z^{2}}$. Their study likely initiated with the observation of the triple (3, 4, 5). There is no academic consensus on the method of generation, purpose, and author. The triples are too large for the method to have been generated by brute force. Its purpose is theorised to be related to number theory, trigonometry, or astronomy. Some historians have considered it a teacher's catalogue of reciprocal pairs.^[16]

Ancient Egyptian arithmetic took an additive approach to multiplication and division. They had a method of representing fractions as sums of distinct unit fractions. The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers.^{[note 1]} Around 300 BC, they knew a formula for the sum of triangular numbers.^{[note 2]} Their awareness of Pythagorean triples is less certain. A problem in the Berlin Papyrus 6610 #1 involves finding integer terms that fulfil a Pythagorean equation, although the solution is a multiple of (3, 4, 5).^[17]

Ancient Greece

Further information: Ancient Greek mathematics

While early civilizations primarily used numbers for concrete practical purposes like commercial activities and tax records, ancient Greek mathematicians began to explore the abstract nature of numbers. Their work survives in the form of book fragments and iterative copies.^[18]

They formalised number theory as a field of study, establishing core concepts such as divisibility, factorisation, the greatest common divisor, and Diophantine equations.^[19] A further contribution was their distinction of various classes of numbers, such as even numbers, odd numbers, and prime numbers.^[20] This included the discovery that numbers for certain geometric lengths are irrational and therefore cannot be expressed as a fraction of integers.^[21]

Most early Greeks did not consider 1 to be a number. They only considered ${\displaystyle 2}$ and above to be numbers. A few scholars in the Greek and later Roman tradition, including Nicomachus, Iamblichus, Boethius, and Cassiodorus, also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider ⁠${\displaystyle 2}$⁠ to be prime either. However, Euclid and a majority of the other Greek mathematicians considered ⁠${\displaystyle 2}$⁠ as prime.^[22]

Archaic to classical

The works of philosophers Thales of Miletus and Pythagoras in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics. Pythagoras founded the school of thought that sought to understand number theory, geometry, astronomy, and music. Plato later reaffirmed the four divisions of the mathematical sciences. Believing that everything revolves around numbers, the Pythagoreans tended to assign them mystical properties. For example, they considered ${\displaystyle 1}$ to represent reason as it is foundational to all other numbers. Further mystics are Philo, Nicomachus, and Iamblichus.^[23]

They visualised numbers as amounts of pebbles and studied polygonal numbers based on their arrangement. This included triangular, square, and pentagonal numbers. They also distinguished between classes such as odd and even, perfect, and amicable numbers. A perfect number is an integer that equals the sum of its proper divisors and was deemed virtuous. Similarly, they considered a pair of distinct integers to be friendly when the sum of the proper divisors of each is equal to the other integer, such as ${\displaystyle 220}$ and ${\displaystyle 284}$.^[24]

Hellenistic to early Roman

Euclid's works shifted towards a rigourous number theory.

Euclid (c. 300 BC) compiled in his Elements contemporary knowledge of geometry and number theory. He incorporated earlier Pythagorean studies of integer properties. In contrast to the Pythagorean mysticism, he used formal proofs to establish mathematical truths and validate theories, a novel feature in ancient Greek mathematics. He ordered his proofs in a logically deductive sequence, ultimately having all results be based on a set of self-evident axioms.^[25]

The earliest surviving records of the study of prime numbers come from the ancient Greeks. Euclid established fundamental results concerning them, such as the infinitude of primes, the fundamental theorem of arithmetic, and the relation between Mersenne primes and perfect numbers. To prove the infinitude, he showed the construction of a larger prime from any arbitrary prime. Euclid's lemma captures an important property of prime numbers and is key for the proof of the fundamental theorem. The fundamental theorem states that every integer greater than 1 is either prime or uniquely composed of a product of prime numbers.^{[note 3][26]} He proved the construction of perfect numbers from prime numbers of the form ${\displaystyle 2^{n}-1}$.^{[note 4][27]} Another contribution to prime numbers was made by Eratosthenes (3rd century BC). He devised a method to identify all primes up to a given bound. The Sieve of Eratosthenes is still used to construct lists of primes.^[28]

Euclid gave the Euclidean algorithm for computing the greatest common divisor of two numbers. This involves repeatedly performing division with remainder and shifting numbers. The method works because the remainder is always less than the divisor. He presented a formula with two parameters to generate all Pythagorean triples and proved their infinitude. And if the greatest common divisor of the parameters is 1, then it generates primitive Pythagorean triples.^[29]

Neopythagorean Nicomachus (2nd century) wrote the first systematic work that focuses solely on number theory entitled Introduction to Arithmetic. His work lacked rigour compared to the Elements and was possibly intended for a wider audience. Statements are typically illustrated with an example but not proved. It mostly discusses integer classes and includes the additional abundant and deficient numbers. In contrast to Euclid, who had a geometric image of numbers, Nicomachus represented numbers using letters with assigned values.^[30]

Late antiquity

In his Arithmetica (transl. 1621), Diophantus explored polynomial equations with positive rational variables.

Diophantus (3rd century) was an influential figure in later Greek arithmetic because of his contributions to number theory and the application of arithmetic operations to algebraic equations. In his Arithmetica, he examined polynomial equations with positive rational variables and experimented with symbolic notation. His work contains problems involving equations of up to 3rd polynomial degree. For example, the Pythagorean equation is a 2nd degree equation with the Pythagorean triples as its solutions. In modern terms, Diophantine analysis studies equations with integer variables, as opposed to positive rational ones.^[31]

An epigram published by Lessing in 1773 appears to be a letter sent by Archimedes to Eratosthenes.^[32] The epigram proposed what has become known as Archimedes's cattle problem; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation). As far as it is known, such equations were first successfully treated by Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

China

Chinese motivation for number theory was to solve problems in astronomy and calendar calculations. This included problems that would later be part of modular arithmetic. An anonymous 4th century tretise titled Sunzi Suanjing contains the first example of the Chinese remainder theorem. An exercise seeks for a number that returns the remainders 2, 3, 2 when divided by 3, 5, 7, respectively. The author gave 23 as the solution but there are infinitely many solutions.^{[33][34][note 5]}

Althought often cited as the first appearance of the theorem, it is a concrete example and not a general theorem [CN]. The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections.^[35] There is also some numerical mysticism in Chinese mathematics,^{[note 6]} but, unlike that of the Pythagoreans, it seems to have led nowhere.

India

Aryabhata devised an algorithm to "pulverise" Diophantine equations.

While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry,^[36] it seems to be the case that Indian mathematics is otherwise an autochthonous tradition;^[37][38] in particular, there is no evidence that Euclid's Elements reached India before the eighteenth century.^[39]

Ancient Indian traditions made inquiry into the nature of numbers. The Vedas contained preference for certain numbers and number classes. The Jains classified the numbers into the subsets of countable, uncountable and infinite numbers. Powers of 10 were assigned individual names. Buddhists and Jains listed terms for powers up to ${\displaystyle 10^{53}}$. The Vedic texts Śulbasūtras concerning the geometry of altar construction details the Pythagorean theorem and Pythagorean triples. Post-classical Indian mathematicians further studied the concepts of irrationality and the number zero. Varāhamihira (6th century) and Brahmagupta (7th century) examined the arithmetic under irrational numbers like the certain square roots. They observed and set rules for the arithmetic treatment of the number zero, which allowed the expansion of the domain of numbers and the development of algebra.^[40]

Āryabhaṭa (476–550 AD) showed that the linear Diophantine equation could be solved by a method called kuṭṭaka (literally pulveriser), which searches for solutions by trial-and-error. The procedure is related to the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Mahāvīra (9th century) refined it by eliminating trial-and-error. He also expressed interest in quadratic equations, number partitions, and fractions.^[41]

Brahmagupta provided a general solution for the linear Diphantine equation. He also gave solutions to specific cases of Pell's equation.^{[note 7]} Later authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was found by Jayadeva (11th century); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (12th century). Further contributions were made by Śrīpati, Narayana Pandita, and Madhava of Sangamagrama.^[42]

RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II

Islamic world

Further information: Mathematics in medieval Islam and Islamic Golden Age

Al-Haytham examined congruences using Wilson's theorem.

In the early ninth century, the caliph al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind, which may^[43] or may not^[44] be Brahmagupta's Brāhmasphuṭasiddhānta). Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī, 953 – c. 1029) builds on it to some extent.^[45] The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.^[46]

Perfect and amicable numbers were also of interest the Arab-speaking world. This is in part due to the importance given to them by Nicomachus's Introduction to Arithmetic.^[47] Thabit ibn Qurra and Abu Mansur al-Baghdadi made notable discoveries. Thabit derived a method for generating amicable numbers and found more pairs. He also translated Nicomachus's Introduction to Arithmetic into Arabic. Analogous to amicable numbers, Abu Mansur discovered a method to obtain balanced numbers. Two integers are called balanced if both sums of the respective proper divisors equal one another.^[48]

Around 1000 AD, Ibn al-Haytham (Alhazen) solved problems involving congruences using what is now called Wilson's theorem, characterizing the prime numbers as the numbers ⁠${\displaystyle n}$⁠ that evenly divide ⁠${\displaystyle (n-1)!+1}$⁠. In his Opuscula, Alhazen considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.^[49] Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit.^[50] in Diophantine analysis, Abu-Mahmud Khujandi stated a special case of Fermat's Last Theorem for n = 3, but his attempted proof of the theorem was incorrect.^[51] Fibonacci took the innovations from Islamic mathematics to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root.^[52]

Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica.^[53]

Early modern

Modern number theory began in the early modern period. The field saw a resurgence in Europe after a period of obscurity. Mathematicians expanded on prior works by the ancient Greeks. The field remained exclusive to elementary number theory, which used elementary methods like proof by infinite descent.^[54]

17th century

Fermat is sometimes called the founder of modern number theory.

Pierre de Fermat (1607–1665) revitalised the study in Western Europe and is sometimes considered the founder of modern number theory. His work is contained in letters to mathematicians and in private marginal notes like in a copy of Diophantus's Arithmetica.^[55] Although he drew inspiration from classical sources, he rarely wrote any proofs.^[56]

One of Fermat's first interests was perfect numbers and amicable numbers. These topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.^[57] In 1638, Fermat claimed that all whole numbers can be expressed as the sum of four squares or fewer.^[58] Fermat also investigated the primality of the Fermat numbers ⁠${\displaystyle 2^{2^{n}}+1}$⁠,^[59] and Marin Mersenne studied the Mersenne primes, prime numbers of the form ${\displaystyle 2^{p}-1}$ with ⁠${\displaystyle p}$⁠ itself a prime.^[60]

In the field of congruences, he described a fundamental result that would become Fermat's little theorem: if a is not divisible by a prime p, then ${\displaystyle a^{p-1}\equiv 1{\bmod {p}}}$. It was later proved by Leibniz and Euler.^[61] Further, if a and b are coprime, then ${\displaystyle a^{2}+b^{2}}$ is not divisible by any prime congruent to −1 modulo 4;^[62] and every prime congruent to 1 modulo 4 can be written in the form ${\displaystyle a^{2}+b^{2}}$.^[63] These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent.^[64]

In Diophantine analysis, Fermat proved by infinite descent that ${\displaystyle x^{4}+y^{4}=z^{4}}$ has no non-trivial solutions in the integers [NONPRIMARY SOURCE]. Fermat also mentioned to his correspondents that ${\displaystyle x^{3}+y^{3}=z^{3}}$ has no non-trivial solutions, and claimed that this could also be proven by infinite descent.^{[65][note 8][66]} He generalised the statements, claiming to have proved that there are no solutions to ${\displaystyle x^{n}+y^{n}=z^{n}}$ for all ${\displaystyle n\geq 3}$. Fermat posed the problem of solving Pell's equation as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker.^[67] Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.

18th century

Euler expanded on the works of Fermat and the ancient Greeks.

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when his correspondent Christian Goldbach pointed him towards some of Fermat's work on the subject.^[68] This has been called the "rebirth" of modern number theory,^[69] after Fermat's relative lack of success in getting his contemporaries' attention for the subject.^[70] Euler's work on number theory includes the following:^[71]

Euler proved many of Fermat's assertions and expanded upon ancient Greek works. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that ${\displaystyle p=x^{2}+y^{2}}$ if and only if ${\displaystyle p\equiv 1{\bmod {4}}}$; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himself^[72]); the lack of non-zero integer solutions to ${\displaystyle x^{4}+y^{4}=z^{2}}$ (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).

In a 1742 letter, Euler received Goldbach's conjecture that every even number is the sum of two primes.^[73] Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes.^[74]

Pell's equation, first misnamed by Euler.^[75] He wrote on the link between continued fractions and Pell's equation.^[76] Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form ${\displaystyle x^{2}+Ny^{2}}$, some of it prefiguring quadratic reciprocity.^[77]

Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1.^[78] In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy.^[79] He did notice there was a connection between Diophantine problems and elliptic integrals,^[79] whose study he had himself initiated.

He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes ⁠${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots }$⁠.^[80] In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.^[81]

Frei 1994

19th century

Legendre pioneered work on the prime number theorem.

At the start of the 19th century, Legendre and Gauss conjectured that as ⁠${\displaystyle x}$⁠ tends to infinity, the number of primes up to ⁠${\displaystyle x}$⁠ is asymptotic to ⁠${\displaystyle x/\log x}$⁠, where ${\displaystyle \log x}$ is the natural logarithm of ⁠${\displaystyle x}$⁠. A weaker consequence of this high density of primes was Bertrand's postulate, that for every ${\displaystyle n>1}$ there is a prime between ⁠${\displaystyle n}$⁠ and ⁠${\displaystyle 2n}$⁠, proved in 1852 by Pafnuty Chebyshev.^[82]

Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations; for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to ${\displaystyle mX^{2}+nY^{2}}$), including defining their equivalence relation, showing how to put them in reduced form, etc.

Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation ${\displaystyle ax^{2}+by^{2}+cz^{2}=0}$^[83] and worked on quadratic forms along the lines later developed fully by Gauss.^[84] In his old age, he was the first to prove Fermat's Last Theorem for ${\displaystyle n=5}$ (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).^[85]

Carl Friedrich Gauss (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The Disquisitiones Arithmeticae (1801), which he wrote three years earlier when he was 21, had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work the law of quadratic reciprocity and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (congruences) and devoted a section to computational matters, including primality tests.^[86] The last section of the Disquisitiones established a link between roots of unity and number theory:

The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.^[87]

In this way, Gauss arguably made forays towards Évariste Galois's work and the area algebraic number theory.

Kleiner 2000

Modern

Dirichlet's novel proof on arithmetic progressions gave rise to analytic number theory.

Starting early in the nineteenth century, the following developments gradually took place:

• The rise to self-consciousness of number theory (or higher arithmetic) as a field of study.^[88]
• The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.
• The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.

Analytic

Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem.^[89] Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.^[90]

Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877),^[91] Proth's theorem (c. 1878),^[92] the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.^[93]

Pafnuty Chebyshev (1821-1894) provided substantiated cases for the PNT such as the existence of primes within specific intervals.

Dirichlet

Kummer

Dedekind

Riemann

Kronecker

Algebraic

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837),^[94] whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.^[95] The first use of analytic ideas in number theory actually goes back to Euler (1730s),^[96] who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point;^[97] Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).^[98]

Geometric

Others

Karam 2000 harvnb error: multiple targets (2×): CITEREFKaram2000 (help)

Contemporary

The conjecture on arithmetic progressions by Erdős (left) has many consequences for open problems. Tao (right) separately coauthored the proof of one of them.

Since 1951 all the largest known primes have been found using these tests on computers.^[a] The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects.^[100]

The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize.

The idea that prime numbers had few applications outside of pure mathematics^[b] was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.^[103]

Number theory was once regarded as the canonical example of pure mathematics with no applications outside the field. In 1970s, it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms. Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors. Elementary number theory is taught in discrete mathematics courses for computer scientists.

The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form.^[104] The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.^[105]

One quirk of number theory is that it deals with statements that are simple to understand but require a high degree of sophistication to solve. Open questions remain, such as the aforementioned Riemann hypothesis that is among the seven Millennium Prize Problems, Goldbach's conjecture on the representation of even numbers as the sum of two primes, and the existence of odd perfect numbers.

Notes

1. Related to Egyptian fractions, primary pseudoperfect numbers were first investigated in 2000.
2. The ${\displaystyle n}$th triangular number ${\displaystyle t_{n}}$ is defined as ${\displaystyle t_{n}=\sum _{k=1}^{n}k}$. Ancient Egyptians discovered the formula ${\displaystyle \sum _{k=1}^{n}t_{n}={\frac {n(n+1)(n+2)}{6}}}$.
3. The fundamental theorem of arithmetic was proved in modern terms by Gauss. It was then extended to abstract algebra, Gaussian integers, Eisenstein integers etc.
4. Open questions remain on the existence of odd perfect numbers and infinitude of Mersenne primes.
5. 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.
6. See, for example, Sunzi Suanjing, Ch. 3, Problem 36, in Lam & Ang 2004, pp. 223–224
7. Pell's equation is a Diophantine equation of the form ${\displaystyle x^{2}-ny^{2}=1}$, where ${\displaystyle n}$ is a positive nonsquare integer.
8. The first known proof was provided by Euler (1753; indeed by infinite descent).

1. A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.^[99]
2. For instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications",^[101] and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful.^[102]

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34. 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.
35. Dauben 2007, p. 310
36. Plofker 2008, p. 119
37. Any early contact between Babylonian and Indian mathematics remains conjectural.
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43. Colebrooke 1817, p. lxv, cited in Hopkins 1990, p. 302. See also the preface in Sachau & Bīrūni 1888 cited in Smith 1958, pp. 168
44. Pingree 1968, pp. 97–125, and Pingree 1970, pp. 103–123, cited in Plofker 2008, p. 256.
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56. Weil 1984, p. 118. This was more so in number theory than in other areas (Mahoney 1994, pp. 283–289). Bachet's own proofs were "ludicrously clumsy" (Weil 1984, p. 33).
57. Mahoney 1994, pp. 48, 53–54. The initial subjects of Fermat's correspondence included proper divisors and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, Tannery & Henry 1891, Vol. II, pp. 72, 74, cited in Mahoney 1994, p. 54.
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63. Tannery & Henry 1891, Vol. II, p. 213.
64. Tannery & Henry 1891, Vol. II, p. 423.
65. Weil 1984, p. 115
66. Weil 1984, pp. 115–116
67. Weil 1984, p. 92.
68. Weil 1984, pp. 2, 172.
69. Weil 1984, pp. 1–2.
70. Weil 1984, p. 2 and Varadarajan 2006, p. 37
71. Varadarajan 2006, p. 39 and Weil 1984, pp. 176–189
72. Weil 1984, pp. 178–179.
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74. Stillwell 2010, p. 40
75. Weil 1984, p. 174. Euler was generous in giving credit to others (Varadarajan 2006, p. 14), not always correctly.
76. Weil 1984, p. 183.
77. Varadarajan 2006, pp. 44–47.
78. Varadarajan 2006, pp. 55–56.
79. Weil 1984, p. 181.
80. Narkiewicz 2000, p. 11
81. Varadarajan 2006, pp. 45–55; see also chapter III.
82. Tchebychev, P. (1852). "Mémoire sur les nombres premiers" (PDF). Journal de mathématiques pures et appliquées. Série 1 (in French): 366–390.. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854
83. Weil 1984, pp. 327–328.
84. Weil 1984, pp. 332–334.
85. Weil 1984, pp. 337–338.
86. Goldstein & Schappacher 2007, p. 14.
87. From the preface of Disquisitiones Arithmeticae; the translation is taken from Goldstein & Schappacher 2007, p. 16
88. 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).
89. Apostol 2000, pp. 1–14
90. Apostol 1976, 7. Dirichlet's Theorem on Primes in Arithmetical Progressions
91. Chabert 2012, p. 261
92. Rosen 2000, p. 342
93. Mollin 2002
94. Apostol 1976, p. 7.
95. See the proof in Davenport & Montgomery 2000, section 1
96. Iwaniec & Kowalski 2004, p. 1.
97. Granville 2008, pp. 322–348.
98. See the comment on the importance of modularity in Iwaniec & Kowalski 2004, p. 1
99. Cooper & Hodges 2016, pp. 37–38
100. • Ziegler 2004
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101. Beiler 1999, p. 2
102. Katz 2004
103. Kraft, James S.; Washington, Lawrence C. (2014). Elementary Number Theory. Textbooks in mathematics. CRC Press. p. 7. ISBN 978-1-4987-0269-0.
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