“Mathematics is the queen of the sciences and number theory is the queen of mathematics.”-Carl Friedrich Gauss
One of the most ancient concepts of mathematics, number theory is a branch of pure mathematics devoted primarily to the study of integers. Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laymen, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background. Until the mid-20th century, number theory was considered as a branch of mathematics having no direct applications to the real world. The advent of digital computers and digital communications revealed that number theory could provide unexpected answers to real-world problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes, testing conjectures, and solving numerical problems once considered out of reach.
Number theory is a broad subject that is classified into:
Analytic number theory:
It may be defined-
- in terms of its tools, as the study of the integers by means of tools from real and complex analysis; or
- in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.
The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.
Elementary number theory: The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.
Algebraic number theory: An algebraic number is any complex number that is a solution to some polynomial equation f(x)= 0 f ( x ) = 0 with rational coefficients; for example, every solution x of x5+(11/2)x3-7x2+9=0 x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 (say) is an algebraic number.
Probabilistic number theory: It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than 0 must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one. At times, a non-rigorous, probabilistic approach leads to a number of heuristic algorithms and open problems, notably Cramér’s conjecture.
Applications in real life:
The keeper of secrets: Number theory was applied to develop increasingly involved algorithms (step-by-step procedures for solving a mathematical problems). In some cryptologic systems, encryption is accomplished by choosing certain prime numbers and then products of those prime numbers as basis for further mathematical operations.
For example, given the plaintext \Hello world” and the key k = 2,
we replace \H” with the letter \F,” \e” with \c,” \l” with \j,” etc.
to produce the ciphertext \Fcjjm umpjb.”
To err is human to correct is the number theory: An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as coding theory. An example of error correcting code is the Reed-Solomon code invented in 1960. The first commercial application in mass-produced consumer products appeared in 1982, with the CD, where two Reed–Solomon codes are used on each track to give even greater redundancy. This is very useful when having to reconstruct the music on a scratched CD.
Other applications include:
- Security System like in banking securities
- E-commerce websites
“Without mathematics, there’s nothing you can do. Everything around you is mathematics. Everything around you is numbers.”-Shakuntala Devi
Contribution of Indian Mathematicians to the theory:
1)Āryabhaṭa (476–550 CE): showed that pairs of simultaneous congruences n≡a1 mod m1,n≡a2 mod m2 could be solved by a method he called kuṭṭaka, or pulveriser; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations.
2)Brahmagupta (628 CE): started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta’s technical terminology. A general procedure (the chakravala, or “cyclic method”) for solving Pell’s equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II’s Bīja-gaṇita (twelfth century).
Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara’s work was translated into English in 1817 by Henry Colebrooke.
3) Srinivasa Ramanujan (1887–1920): The Indian mathematical genius made remarkable contributions in several areas of mathematics, including Number Theory. He revolutionalized the study of some areas of number theory by making great contributions. For example, Theory of Partitions, Ramanujan’s tau function, The Rogers-Ramanujan Continued Fractions, and so on. Most of his research work on Number Theory arose out of q-series and theta functions. He developed his own theory of elliptic functions, and applied his theory to develop some truly different areas, like, hypergeometric-like series for 1/pi, class invariants, continued fractions and many more.
The theory of numbers, then, is a vast and challenging subject as old as mathematics and as fresh as today’s news. Its problems retain their fascination because of an apparent (often deceptive) simplicity and an irresistible beauty. With such a rich and colorful history, number theory surely deserves to be called, in the famous words of Gauss, “the queen of mathematics.”