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Mathematical finance

Mathematical finance is the branch of applied mathematics concerned with the financial markets.

The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock.

In terms of practice, mathematical finance also overlaps heavily with the fields of financial engineering and computational finance. Arguably, all three are largely synonymous, although the latter two focus on application, while the former focuses on modelling and derivation; see Quantitative analyst.

Many universities around the world now offer degree and research programs in mathematical finance.

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Mathematical economics

The term mathematical economics is employed in two main senses:

(1) As a specialized area of study, mathematical economics is a distinct sub-field within the discipline of economics concerned with the application and development of mathematical techniques to shed light on economic problems. Paul Samuelson's Foundations of Economic Analysis (1947) is considered a classic statement of contemporary mathematical economics.

(2) As a general set of analytical methods, mathematical economics—or, to distinguish it from the first sense employed above, the mathematical method of economics—represents a widely though by no means universally adopted approach to the presentation and interpretation of economic problems.
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While the field of mathematical economics is widely acclaimed (due in large part to the success of its progeny, mathematical finance), the widespread use of mathematical methods in economics is controversial. Opponents of the mathematical method, notably the Austrian School, argue that the use of formal techniques lends to the field an impression of scientific exactness that, by nature of the eccentricities of its human subject matter, is unfeasible, even in principle. By contrast, proponents argue that the validity of the mathematical method derives from economists' distinctive assumptions about the internal mechanics of economic decision-making: economic agents are generally (and, to many social scientists, strangely) assumed to be (i) rational and (ii) self-interested, from which it follows that an economic agent's deductions and behaviour may be compared against the calculations reached using formal logical and analytical techniques, including optimization and other advanced mathematical procedures. The rational-actor framework has been disputed as a valid characterization of human decision-making, but it remains the primary framework in mainstream economics.

Although the mathematical method of economics has evolved through geometric, algebraic and higher forms, a solid grasp of modern algebraic methods is a prerequisite for formal study, not only in mathematical economics, but in economics generally. For instance, the Journal of Economic Theory, one of the most prominent academic references in the field, is the apotheosis of the mathematical approach—though surprisingly, according to the Editors, it is a non-specialist journal.




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Arithmetic

History
It is the method known as the "Method of the Indians" or in Latin "Modus Indoram" that has become our arithmetic today. Prior to this, basic arithmetic operations were highly complicated affairs. Seventh century Syriac Bishop Severus Sebhokt mentioned this method and stated that the method of the Indians is beyond description. Indian arithmetic was much simpler than the Greek arithmetic simply due to the simplicity of the Indian number system which had a zero and place value notation. Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci or Leonardo of Pisa is one of the first European mathematicians who introduced the "Method of the Indians" to Europe. In his famous book "Liber Abaci" Fibonacci says that compared to this new method all other methods were mistakes.


The prehistory of arithmetic is limited by a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango Bone from Africa, dating from 18,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 BC, although historians can only infer the methods utilized to generate the arithmetical results (see Plimpton 322). Likewise, definitive addition, subtraction, multiplication, and division facts are used within the unit fraction system, which can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 BC, copied from 1850 BC (Mathematathe septem liberales artes (seven liberal arts).

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu- Arabic numeral based arithmetic was developed by great Indian mathematicians Aryabhatta, Brahmagupta and Bhaskara. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.




Further information: Introduction to Arithmetic

[edit] Decimal arithmetic
Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of 0 as a number comparable to the other basic digits.

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,102,10,1,10-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.


[edit] Arithmetic operations
The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field.


[edit] Addition (+)
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0.


[edit] Subtraction (−)
Main article: Subtraction
Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.


[edit] Multiplication (× or ·)
Main article: Multiplication
Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both are simply called factors.

Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1.


[edit] Division (÷ or /)
Main article: Division (mathematics)
Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers and negative one). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it will obey all the properties of multiplication.

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Analytic number theory

Analytic number theory is the branch of number theory that uses methods from mathematical analysis. Its first major success was Dirichlet's application of analysis to prove Dirichlet's theorem on arithmetic progressions, stating the existence of infinitely many primes in arithmetic progressions of the form a + nb, where a and b are relatively prime. The proofs of the prime number theorem based on the Riemann zeta function are another milestone.

The outline of the subject remains similar to the heyday of the subject in the 1930s. Multiplicative number theory deals with the distribution of the prime numbers, applying Dirichlet series as generating functions. It is assumed that the methods will eventually apply to the general L-function, though that theory is still largely conjectural. Additive number theory has as typical problems Goldbach's conjecture and Waring's problem.

Methods have changed somewhat. The circle method of

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Algebraic number theory

Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. An algebraic number field is any finite (and therefore algebraic) field extension of the rational numbers. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed — Galois theory, group cohomology, class field theory, group representations and L-functions — is that it allows one to recover that order partly for this new class of numbers.

Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.
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Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.

Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics).

The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.


elementary number theory

In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler's φ function, integer sequences, factorials and Fibonacci numbers all also fall into this area.

Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include:

* The Goldbach conjecture concerning the expression of even numbers as sums of two primes.
* Catalan's conjecture (now Mihăilescu's theorem) regarding successive integer powers.
* The twin prime conjecture about the infinitude of prime pairs.
* The Collatz conjecture concerning a simple iteration.
* Fermat's last theorem (stated in 1637, but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xn + yn = zn for some integer n greater than 2.

The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem).

Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem (PNT) and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one.

Algebraic number theory

In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed—Galois theory, group cohomology, class field theory, group representations and L-functions—is that it allows to recover that order partly for this new class of numbers.

Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.

Geometric number theory

Geometric number theory (traditionally called geometry of numbers) incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings.

Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.

Computational number theory

Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.

History

Vedic number theory

Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era. The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras, which were written between the 8th and 6th centuries BC. Baudhayana (c. 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also used simultaneous Diophantine equations with up to four unknowns. Apastamba (c. 600 BC) used simultaneous Diophantine equations with up to five unknow

Jaina number theory

In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE. The Jaina text Surya Prajinapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders:

* Enumerable: lowest, intermediate and highest.
* Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
* Infinite: nearly infinite, truly infinite, infinitely infinite.

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).

The highest enumerable number N of the Jains corresponds to the modern concept of aleph-null \aleph_0 (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of transfinite cardinal numbers, of which \aleph_0 is the smallest.

In the Jaina work on the theory of sets, two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.

Hellenistic number theory

Number theory was a favorite study among the Hellenistic mathematicians of Alexandria, Egypt from the 3rd century CE, who were aware of the Diophantine equation concept in numerous special cases. The first Hellenistic mathematician to study these equations was Diophantus.

Diophantus also looked for a method of finding integer solutions to linear indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.

Classical Indian number theory

Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations. Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. This kuttaka algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy. He also found the general solution to the indeterminate linear equation using this method.

Brahmagupta in 628 handled more difficult Diophantine equations. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x2 + 1 = y2. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The equation 61x2 + 1 = y2 was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. The general solution to this particular form of Pell's equation was found over 70 years later by Leonhard Euler, while the general solution to Pell's equation was found over 100 years later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations. Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later. Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic and higher-order polynomial equations. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations.

Islamic number theory

From the 9th century, Islamic mathematicians had a keen interest in number theory. The first of these mathematicians was the Arab mathematician Thabit ibn Qurra, who discovered a theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. In the 10th century, Al-Baghdadi looked at a slight variant of Thabit ibn Qurra's theorem.

In the 10th century, al-Haitham seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k − 1(2k − 1) where 2k − 1 is prime. Al-Haytham is also the first person to state Wilson's theorem, namely that if p is prime then 1 + (p − 1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771.

Amicable numbers played a large role in Islamic mathematics. In the 13th century, Persian mathematician Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 still many years before Euler's contribution.

Early European number theory

Number theory began in Europe in the 16th and 17th centuries, with François Viète, Bachet de Meziriac, and especially Fermat, whose infinite descent method was the first general proof of diophantine questions. Fermat's last theorem was posed as a problem in 1637, a proof of which wasn't found until 1994. Fermat also posed the equation 61x2 + 1 = y2 as a problem in 1657.

In the eighteenth century, Euler and Lagrange made important contributions to number theory. Euler did some work on analytic number theory, and found a general solution to the equation 61x2 + 1 = y2, which Fermat posed as a problem. Lagrange found a solution to the more general Pell's equation. Euler and Lagrange solved these Pell equations by means of continued fractions, though this was more difficult than the Indian chakravala method.

Beginnings of modern number theory

Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put together the first systematic theories in Europe. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers.

The formulation of the theory of congruences starts with Gauss's Disquisitiones. He introduced the symbolism

a \equiv b \pmod c,

and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularised it.

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. The following have also contributed to the subject: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker. The theory extends to include cubic and biquadratic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer).

To Gauss is also due the representation of numbers by binary quadratic forms.

Prime number theory

A recurring and productive theme in number theory is the study of the distribution of prime numbers. Carl Friedrich Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager.

Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Riemann introduced complex analysis into the theory of the Riemann zeta function. This led to a relation between the zeros of the zeta function and the distribution of primes, eventually leading to a proof of prime number theorem independently by Hadamard and de la Vallée Poussin in 1896. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949+. Here elementary means that it does not use techniques of complex analysis; however, the proof is still very ingenious and difficult. The Riemann hypothesis, which would give much more accurate information, is still an open question.

Nineteenth-century developments

Cauchy, Poinsot (1845), Lebesgue(?) (1859, 1868), and notably Hermite have added to the subject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith.

Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's last theorem:

x^n+y^n \neq z^n, (x,y,z \neq 0, n > 2)

which Euler and Legendre had proven for n = 3,4 (and therefore by implication, all multiples of 3 and 4), Dirichlet showing that x^5+y^5 \neq az^5. Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany were Kronecker, Kummer, Schering, Bachmann, and Dedekind. In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory.

Twentieth-century developments

Major figures in twentieth-century number theory include Paul Erdős, Gerd Faltings, G. H. Hardy, Edmund Landau, John Edensor Littlewood, Srinivasa Ramanujan and André Weil.

Milestones in twentieth-century number theory include the proof of Fermat's Last Theorem by Andrew Wiles in 1994 and the proof of the related Taniyama–Shimura conjecture in 1999.

Quotations

* Mathematics is the queen of the sciences and number theory is the queen of mathematics. — Gauss
* God invented the integers; all else is the work of man. — Kronecker
* I know numbers are beautiful. If they aren't beautiful, nothing is. — Erdős


References

* Apostol, T. M. (1986). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9.
* Dedekind, Richard (1963). Essays on the Theory of Numbers. Cambridge University Press. ISBN 0-486-21010-3.
* Davenport, Harold (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.). Cambridge University Press. ISBN 0-521-63446-6.
* Guy, Richard K. (1981). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-90593-6.
* Hardy, G. H. and Wright, E. M. (1980). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. ISBN 0-19-853171-0.
* Niven, Ivans Herbert S. Zuckermans and Hugh L. Montgomery (1991). An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books. ISBN 0-471-62546-9.
* Ore, Oystein (1948). Number Theory and Its History. Dover Publications, Inc.. ISBN 0-486-65620-9.
* Smith, David. History of Modern Mathematics (1906) (adapted public domain text)
* Dutta, Amartya Kumar (2002). 'Diophantine equations: The Kuttaka', Resonance - Journal of Science Education.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Arabic/Islamic mathematics', MacTutor History of Mathematics archive.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Index of Ancient Indian mathematics', MacTutor History of Mathematics archive.
* O'Connor, John J. and Robertson, Edmund F. (2004). 'Numbers and Number Theory Index', MacTutor History of Mathematics archive.
* Important publications in number theory

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