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Number theory is one of the oldest parts of mathematics, but modern number theory is not a single ancient subject carried forward unchanged. It is a layered discipline....

Divisor functions measure the positive divisors of an integer. They are among the first examples of arithmetic functions, because their values depend directly on the prime...

Modular arithmetic is not only a theoretical language for divisibility. It is also one of the main tools of computation with integers.

Let

Number theory studies arithmetic simultaneously at two levels:

Quadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers.

One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.

Modular arithmetic often requires computing powers such as

Computation has become an essential part of number theory. Classical arithmetic relied mainly on symbolic reasoning and hand calculations. Modern arithmetic combines rigorous...

The Prime Number Theorem describes the average distribution of primes up to a large number $x$:

Classically, number theory studied special analytic functions such as modular forms. These functions satisfy strong symmetry conditions under actions of arithmetic groups.

The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.

The Chinese remainder theorem describes when several congruence conditions can be combined into one congruence. Its cleanest form occurs when the moduli are pairwise coprime.

The Prime Number Theorem states that

Representation theory studies abstract algebraic objects by expressing them as linear transformations of vector spaces.

The real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and...

Gauss sums arise from combining multiplicative and additive structures modulo a prime. They form one of the fundamental tools of analytic and algebraic number theory.

Category theory studies mathematical structures through objects and maps between them. Instead of looking only at what objects are made of, it studies how they relate to other...

One of the central problems in arithmetic geometry is understanding the number of solutions of polynomial equations over finite fields.

The ordinary absolute value on the real numbers measures magnitude:

The theory of quadratic residues asks a fundamental question:

A system of congruences asks for an integer satisfying several congruence conditions simultaneously.

In ordinary arithmetic, division by a nonzero number means multiplication by its reciprocal. Modular arithmetic is more delicate. A residue class may or may not have a...

A linear congruence is a congruence of the form

Arithmetic modulo $n$ is arithmetic performed on residue classes modulo $n$. Instead of distinguishing all integers separately, we identify integers that have the same...

Congruence modulo $n$ groups integers according to their remainders after division by $n$. If two integers have the same remainder, they are congruent modulo $n$.

Ordinary equality compares integers exactly. In many arithmetic problems, however, only the remainder after division matters.

The infinitude of primes guarantees that primes continue indefinitely, but it says nothing about how frequently primes occur.

Euclid proved that there are infinitely many primes by contradiction. Euler discovered a very different proof based on infinite series and products.

Euclid's proof of the infinitude of primes is one of the earliest examples of a general argument in number theory. It does not depend on computation, experimentation, or...

Prime numbers are the building blocks of the positive integers. Once unique prime factorization is known, a natural question arises: are there only finitely many primes, or do...

An arithmetic function is a function whose domain is the positive integers. It assigns a value to each integer

Unique prime factorization says that every integer $n>1$ can be written as a product of primes. The canonical prime decomposition is the ordered and exponentiated version of...

A vector space over a field $F$ is a set $V$ equipped with addition and scalar multiplication satisfying the usual algebraic rules.

The logarithmic integral is the function

Classical topology studies geometric spaces using invariants such as homology and cohomology. Over the complex numbers, algebraic varieties can often be viewed as topological...

One of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field.

Measure theory extends the ideas of length, area, volume, and integration to more general settings. In number theory, measure appears in probability, harmonic analysis,...

The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that

Euler criterion gives an efficient way to decide whether an integer is a square modulo an odd prime. Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$....

Arithmetic geometry often studies families of algebraic curves varying over arithmetic bases. The most important base is

One of the most important classes of number fields arises from the solutions of the equation

The prime counting function

The Legendre symbol

Let $p$ be an odd prime and let $a\in\mathbb{Z}$. The Legendre symbol is defined by

A quadratic congruence is a congruence involving a square. The basic form is

Topology studies continuity, convergence, connectedness, and geometric structure in an abstract setting. In number theory, topology appears naturally in real analysis, complex...

An algebraic curve is a geometric object whose dimension is one. Curves are among the oldest and most important objects in number theory and algebraic geometry.

The familiar fields

One of the oldest questions in number theory asks how prime numbers are distributed among the positive integers. Since primes become less frequent as numbers grow larger,...

The fundamental theorem of arithmetic states that every integer $n>1$ can be written as a product of prime numbers, and that this product is unique up to the order of the factors.

A Diophantine equation is first an arithmetic object. It asks for solutions in integers or rational numbers. But every polynomial equation also defines a geometric object.

Two integers $a$ and $b$, not both zero, are called coprime if their greatest common divisor is $1$:

Let $a$ and $b$ be integers. An integer of the form

The real numbers $\mathbb{R}$ extend the rational numbers $\mathbb{Q}$ by filling gaps such as

Geometry is not only concerned with spaces themselves, but also with maps between spaces. In algebraic geometry and arithmetic geometry, these maps are called morphisms.

A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider

In analytic number theory, one often studies sums of the form

A central problem in number theory is to study solutions of polynomial equations whose coordinates belong to a specified number system. Two important cases are:

The Euclidean algorithm computes the greatest common divisor of two integers. The extended Euclidean algorithm does more. It also expresses the gcd as an integer linear...

The greatest common divisor of two integers can be found by listing divisors, but this method becomes inefficient for large numbers. For example, finding

An exponential Diophantine equation is a Diophantine equation in which one or more unknowns appear as exponents. Typical examples include

A Catalan-type equation is a Diophantine equation involving powers whose values differ by a small amount. The classical example is

Let $a$ and $b$ be nonzero integers. An integer $m$ is called a common multiple of $a$ and $b$ if

Analytic number theory studies infinite sums, products, and integrals. Before such expressions can be manipulated safely, one must understand the meaning of convergence.

One of the oldest questions in number theory asks which integers can be written as sums of squares. Typical examples are

Let $a$ and $b$ be integers, not both zero. An integer $d$ is called a common divisor of $a$ and $b$ if

Abstract algebra studies sets equipped with operations. In number theory, these structures organize arithmetic behavior.

Euler products arise when an infinite series has coefficients controlled by multiplication. The simplest and most important example is the zeta series

A Pell equation is a Diophantine equation of the form

The division algorithm is one of the basic structural facts about the integers. It says that any integer can be divided by a positive integer with a unique quotient and remainder.

Classical algebraic geometry studies varieties defined by polynomial equations. This theory works well over algebraically closed fields, especially over $\mathbb{C}$. However,...

A central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial

An infinite product has the form

A Pythagorean triple is a triple of positive integers

A positive integer $n>1$ is called composite if it is not prime.

A mathematical proof is a logically complete argument establishing the truth of a statement from accepted assumptions, definitions, and previously proved results.

Prime numbers are the fundamental building blocks of arithmetic.

A set is a collection of objects, called its elements. If $x$ is an element of a set $A$, we write $x \in A$. If $x$ is not an element of $A$, we write $x \notin A$.

A clear explanation of the Long Pressed Name problem using a two-pointer scan.

A clear explanation of counting good starting indices using next-jump preprocessing and dynamic programming.

A clear explanation of the Rectangle Area II problem using sweep line and merged active y-intervals.

A clear explanation of counting subarrays whose sum is divisible by k using prefix sums and remainder frequencies.

A clear explanation of returning the k closest points to the origin using squared distance and sorting.

A clear explanation of the Maximize Distance to Closest Person problem using gaps between occupied seats.

A clear explanation of the Shifting Letters problem using suffix sums and modulo arithmetic.

A clear explanation of comparing rational numbers written as decimal strings with optional repeating parts.

A clear explanation of the Shortest Path Visiting All Nodes problem using multi-source BFS and bitmask state compression.

A clear explanation of minimizing malware spread by analyzing connected components with Union Find.

A clear explanation of matching a binary tree preorder traversal by greedily flipping nodes.

A clear explanation of the Hand of Straights problem using sorting, frequency counting, and greedy grouping.

A clear explanation of counting index triplets with duplicate values using frequency counts and combinatorics.

A clear explanation of designing an iterator over a run-length encoded sequence without expanding it.

A clear explanation of generating all powerful integers using bounded powers and a set.

A clear explanation of placing even numbers at even indices and odd numbers at odd indices using two pointers.

A clear explanation of the Longest Mountain in Array problem using peak detection and two-pointer expansion.

A clear explanation of finding the lexicographically smallest string after queue operations using rotation and sorting.

A clear explanation of sorting an array using prefix reversals by repeatedly placing the largest remaining value.

A clear explanation of merging consecutive stone piles with minimum cost using interval dynamic programming.

A clear explanation of finding the minimum banana-eating speed using binary search on the answer.