Number Theory

Number theory is the branch of mathematics concerned primarily with the properties of integers and the arithmetic structures arising from them. Its central subjects include divisibility, prime numbers, congruences, and equations whose solutions must be integers or rational numbers. Modern number theory also studies extensions of the rational numbers and connects arithmetic questions with analysis, algebra, geometry, and computation. These approaches overlap rather than forming sharply separated disciplines. (math.mit.edu)

Divisibility and prime numbers

Divisibility provides the elementary framework of number theory. An integer $a$ divides an integer $b$, written $a\mid b$, if $b=ak$ for some integer $k$. The greatest common divisor of two integers measures their shared factors; the Euclidean algorithm computes it by repeated division with remainder. This procedure is also a foundation for solving linear equations in integers. (ocw.mit.edu)

A prime is a positive integer greater than one whose only positive divisors are one and itself. The fundamental theorem of arithmetic states that every integer greater than one has a prime factorization unique apart from the ordering of factors. Primes therefore serve as the multiplicative building blocks of positive integers. Euclid proved that infinitely many primes exist: any proposed finite list can be used to construct an integer having a prime divisor outside that list. (math.mit.edu)

Elementary number theory includes much more than factorization. It studies arithmetic functions, representations as sums of squares, continued fractions, and quadratic reciprocity. Here “elementary” describes methods that avoid certain advanced machinery, especially complex analysis; it does not imply that the questions or proofs are easy. (math.mit.edu)

Congruences and arithmetic equations

Modular arithmetic organizes integers according to their remainders. For a positive integer $m$,

means that $m\mid(a-b)$. Addition and multiplication preserve congruence, allowing arithmetic calculations to take place within finitely many residue classes. For example, $17\equiv5\pmod{12}$. The Chinese remainder theorem states that prescribed remainders modulo pairwise coprime positive integers determine a unique residue class modulo their product. (ocw.mit.edu)

A Diophantine equation is an equation for which integer solutions are sought; related problems seek rational solutions. Typical questions ask whether solutions exist, whether there are finitely or infinitely many, and how all solutions can be described. Examples include $ax+by=c$, Pythagorean triples satisfying $x^2+y^2=z^2$, and Pell-type equations $x^2-Dy^2=1$. Unlike unrestricted equation solving, these problems depend decisively on the permitted number system. (ocw.mit.edu)

Congruences can exclude possible solutions. Every square is congruent to zero or one modulo four, so $x^2+y^2=3$ has no integer solutions. Passing modular tests, however, is not generally sufficient to establish a global solution. This distinction between arithmetic modulo individual primes and arithmetic over the integers motivates local and global methods. (ocw.mit.edu)

Algebraic, analytic, and geometric methods

Algebraic number theory extends integer arithmetic to larger systems. A number field is a finite extension of the rational-number field, and its algebraic integers satisfy monic polynomial equations with integer coefficients. Unique factorization of elements can fail in these systems. The study of ideals, prime ideals, units, and ideal class groups supplies a framework for understanding that failure and recovering useful factorization structures. (math.mit.edu)

Analytic number theory uses analysis, including complex analysis, to investigate arithmetic patterns. Its central results include the prime number theorem:

where $\pi(x)$ counts primes not exceeding $x$, and the symbol $\sim$ means that the ratio approaches one as $x$ increases without bound. Thus primes become less frequent in an average, precisely quantifiable sense. The theorem was proved independently by Jacques Hadamard and Charles-Jean de la Vallée Poussin in 1896. (claymath.org)

The Riemann zeta function links prime factorization with analytic behavior. More generally, zeta functions and L-functions encode arithmetic information in functions whose values, poles, and zeros can be studied analytically. Algebraic geometry provides another viewpoint: rational solutions become rational points on curves and higher-dimensional spaces. Elliptic curves are particularly important examples, combining geometric equations with a group structure on their points. (claymath.org)

Historical development and major problems

Euclid’s work established foundational results about primes and divisibility. In 1801, Carl Friedrich Gauss published Disquisitiones Arithmeticae, a systematic treatment that organized congruences, quadratic reciprocity, and quadratic forms. Bernhard Riemann introduced influential analytic ideas concerning prime distribution in his 1859 paper. (math.mit.edu)

Fermat’s Last Theorem states that $x^n+y^n=z^n$ has no positive integer solutions when the integer exponent $n$ exceeds two. Andrew Wiles’s proof, published in 1995 alongside supporting work with Richard Taylor, demonstrated how an elementary-looking arithmetic statement could depend on deep relationships between elliptic curves and modular forms. (annals.math.princeton.edu)

The Riemann hypothesis remains unresolved. It asserts that every nontrivial zero of the analytically continued zeta function has real part $1/2$. Its significance includes strong consequences for the error in estimates of prime distribution; checking many zeros computationally does not establish the assertion for all zeros. (claymath.org)

Computation and applications

Computational number theory develops algorithms for primality testing, factorization, modular calculations, and arithmetic on elliptic curves. It distinguishes deciding whether an integer is prime from finding the factors of a composite integer. Applications in cryptography exploit arithmetic operations that are efficient to perform while particular inverse problems are believed difficult at suitable sizes. RSA uses arithmetic related to integer factorization, while elliptic-curve systems use group operations and discrete logarithm problems. Elliptic curves also support factorization algorithms and methods that produce verifiable primality proofs. (ocw.mit.edu)