Euler Products
Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes.
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Number-Theory
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Dirichlet Series
An arithmetic function $fn$ can be encoded into an infinite series of the form
Average Orders of Arithmetic Functions
Arithmetic functions often fluctuate strongly from one integer to the next.
Möbius Inversion
Many arithmetic functions are defined through sums over divisors. For example,
Dirichlet Convolution
Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility.
Completely Multiplicative Functions
An arithmetic function is a function defined on the positive integers. Such a function
Frobenius Automorphisms
One of the deepest ideas in algebraic number theory is that prime numbers possess hidden symmetry inside field extensions.
Parity Problem
Sieve methods are extremely effective for estimating how many integers avoid small prime factors. They have produced major results about:
Appendix A.1 Sets and Functions
A set is a collection of objects called elements.
Liouville Function
The Liouville function is an arithmetic function denoted by
Decomposition and Inertia Groups
Let
Chen's Theorem
The Twin Prime Conjecture states that infinitely many primes satisfy
Euler Totient Function
Euler's totient function is an arithmetic function denoted by
Ramification of Primes
In the ordinary integers, every nonzero integer factors uniquely into prime numbers.
Bombieri-Vinogradov Theorem
The Prime Number Theorem for arithmetic progressions states that for
The Langlands Program
The Langlands program is one of the largest and most influential research programs in modern mathematics.
Möbius Function
The Möbius function is an arithmetic function denoted by
Local Fields
Classical number theory studies arithmetic globally over fields such as
Brun-Titchmarsh Theorem
One of the central problems of analytic number theory is understanding how primes distribute among residue classes.
The Birch and Swinnerton-Dyer Conjecture
An elliptic curve over $\mathbb{Q}$ may be written in Weierstrass form
Divisor Functions
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...
$p$-Adic Numbers
The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic...
Large Sieve
Classical sieve methods estimate how many integers survive congruence restrictions. The large sieve approaches these problems from a different direction.
The Riemann Hypothesis
The Riemann zeta function is one of the central objects in mathematics.
Applications to Computation
Modular arithmetic is not only a theoretical language for divisibility. It is also one of the main tools of computation with integers.
Valuations and Absolute Values
In ordinary analysis, the absolute value
Selberg Sieve
Brun's sieve introduced the idea of estimating sifted sets through truncated inclusion-exclusion. However, Brun's method often produced bounds that were technically difficult...
Fermat's Last Theorem
Fermat's Last Theorem states that there are no positive integers
Fast Modular Exponentiation
Modular arithmetic often requires computing powers such as
Dedekind Domains
Ordinary integers satisfy several remarkable properties simultaneously:
Brun Sieve
Sieve methods are techniques for counting integers that remain after removing residue classes modulo primes.
Geometric Langlands Theory
The classical Langlands program relates:
Open Problems in Number Theory
Number theory contains some of the oldest and deepest unsolved problems in mathematics.
Chinese Remainder Theorem
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.
Principal Ideals
Let $R$ be a commutative ring. An ideal $I\subseteq R$ is called principal if there exists an element $\alpha\in R$ such that
Schnirelmann Density
In additive number theory, ordinary asymptotic density is often too weak to control additive behavior.
Shimura Varieties
Modular curves parameterize elliptic curves and connect modular forms with arithmetic geometry.
Arithmetic Statistics
Arithmetic statistics studies the distribution of arithmetic objects inside large families.
Systems of Congruences
A system of congruences asks for an integer satisfying several congruence conditions simultaneously.
Discriminants
The discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic.
Additive Bases
A central question in additive number theory asks whether every integer can be represented as a sum of elements from a fixed set.
Trace Formulas
Fourier analysis decomposes functions into harmonic frequencies.
Probabilistic Models for Primes
Prime numbers are deterministic objects, but many aspects of their distribution resemble random behavior.
Modular Inverses
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...
Units and Dirichlet Unit Theorem
Let $K$ be a number field and let
Exponential Sums
Exponential sums are among the central tools of analytic number theory.
Automorphic $L$-Functions
The Riemann zeta function
Random Matrices and Zeta Zeros
The Riemann zeta function is defined for $\operatorname{Re}s>1$ by
Linear Congruences
A linear congruence is a congruence of the form
Class Groups
In ordinary integers, every ideal is generated by a single element:
Circle Method
Many problems in additive number theory ask whether an integer can be represented in the form
Functoriality
The Langlands program predicts that many different arithmetic objects are connected by systematic transfers.
Probabilistic Algorithms
A probabilistic algorithm uses random choices during its execution. In number theory, this is often a practical advantage rather than a weakness.
Arithmetic Modulo $n$
Arithmetic modulo $n$ is arithmetic performed on residue classes modulo $n$. Instead of distinguishing all integers separately, we identify integers that have the same...
Ideals and Prime Ideals
In ordinary integers, every number factors uniquely into primes. In many rings of algebraic integers, this property fails.
Waring's Problem
Waring's problem asks whether every sufficiently large positive integer can be written as a sum of a bounded number of fixed powers.
Galois Representations
Galois groups encode the symmetries of algebraic equations and field extensions.
Probabilistic Primality
A primality test determines whether an integer is prime.
Residue Classes
Congruence modulo $n$ groups integers according to their remainders after division by $n$. If two integers have the same remainder, they are congruent modulo $n$.
Unique Factorization Failure
One of the central properties of the ordinary integers is unique factorization.
Goldbach Problems
Goldbach-type problems ask whether integers can be represented as sums of primes. They are among the oldest and most famous problems in additive number theory.
The Langlands Program
The Langlands program is one of the most ambitious and influential theories in modern mathematics.
Smooth Numbers
A positive integer is called $y$-smooth if all of its prime factors are at most $y$.
Congruence Relations
Ordinary equality compares integers exactly. In many arithmetic problems, however, only the remainder after division matters.
Norm and Trace
Let $K$ be a number field of degree
Sumsets
Additive number theory studies arithmetic structure through addition of integers and subsets of integers.
Automorphic Representations
Classical modular form theory begins with analytic functions satisfying symmetry conditions.
Random Integers
Number theory often studies exact statements about individual integers. For example, one may ask whether a given integer is prime, squarefree, smooth, or representable as a...
Distribution Heuristics of Primes
The infinitude of primes guarantees that primes continue indefinitely, but it says nothing about how frequently primes occur.
Ring of Integers
In ordinary arithmetic, the integers
Generalized Riemann Hypothesis
The classical Riemann Hypothesis concerns the zeros of the Riemann zeta function
Automorphic Forms
Modular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group
Zero-Knowledge Proofs
A zero-knowledge proof allows one party to convince another that a statement is true without revealing why it is true.
Euler's Proof
Euclid proved that there are infinitely many primes by contradiction. Euler discovered a very different proof based on infinite series and products.
Number Fields
A number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying
Nonvanishing Results
A central theme in analytic number theory is determining when an $L$-function is nonzero at a particular point.
The Modularity Theorem
For centuries, elliptic curves and modular forms were studied as separate objects.
Post-Quantum Cryptography
Modern public-key cryptography relies heavily on two computational assumptions:
Euclid's Proof
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...
Minimal Polynomials
An algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a...
Primes in Arithmetic Progressions
An arithmetic progression is a sequence of the form
Elliptic Curves and Modularity
An elliptic curve is simultaneously:
Lattice Cryptography
Lattice cryptography is a family of cryptographic systems based on the presumed hardness of computational problems on high-dimensional lattices.
Infinitude of Primes
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...
Algebraic Integers
The ordinary integers
Dirichlet $L$-Functions
The Riemann zeta function
Modular Curves
The modular group acts on the upper half-plane by fractional linear transformations:
Pairing-Based Cryptography
Pairing-based cryptography uses special maps defined on elliptic curve groups. A pairing is a function
Arithmetic Functions from Factorization
An arithmetic function is a function whose domain is the positive integers. It assigns a value to each integer
Diophantine Approximation
Diophantine approximation studies how closely real numbers can be approximated by rational numbers.
Orthogonality Relations
Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes...
Hecke Operators
Modular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators.
Elliptic Curve Cryptography
Elliptic curve cryptography is a public-key cryptographic framework based on the arithmetic of elliptic curves over finite fields.
Canonical Prime Decomposition
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...
Pell Equations via Continued Fractions
Recall that a Pell equation has the form
Dirichlet Characters
The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as
Cusp Forms
Modular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass.
Diffie-Hellman Key Exchange
Secure communication requires two parties to share secret information. In classical symmetric cryptography, both parties must already possess the same secret key before...
Unique Prime Factorization
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.
Convergents
The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.
Connections with Prime Distribution
The Riemann zeta function was introduced through the series
Eisenstein Series
Among all modular forms, Eisenstein series are the most explicit and computationally accessible.
RSA Cryptosystem
Classical cryptography uses a shared secret key. Both sender and receiver must know the same secret information in advance.
Coprime Integers
Two integers $a$ and $b$, not both zero, are called coprime if their greatest common divisor is $1$:
Rational Approximations
Many important numbers are irrational:
Explicit Formulae
One of the deepest ideas in analytic number theory is that the zeros of the zeta function determine the distribution of prime numbers.
Modular Forms
Modular forms are among the central objects of modern number theory.
Symbolic and Numeric Computation
Modern number theory relies heavily on computation. Two broad computational paradigms dominate the subject:
Bezout Identities
Let $a$ and $b$ be integers. An integer of the form
Infinite Continued Fractions
Finite continued fractions correspond exactly to rational numbers. When the Euclidean algorithm never terminates, the continued fraction becomes infinite.
Riemann Hypothesis
The Riemann zeta function has nontrivial zeros inside the critical strip
Modular Functions
The modular group acts on the upper half-plane by fractional linear transformations:
Algorithms for Elliptic Curves
Elliptic curves occupy a central position in modern number theory, arithmetic geometry, and cryptography.
Extended Euclidean Algorithm
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...
Finite Continued Fractions
A finite continued fraction is an expression of the form
Zeros of the Zeta Function
The zeros of the Riemann zeta function are the complex numbers $s$ satisfying
Modular Groups
Modular forms begin with the action of certain matrix groups on the complex upper half-plane.
Algorithms for Modular Forms
Modular forms are highly structured analytic functions with deep arithmetic properties. Although their definitions involve complex analysis and group actions, modular forms...
Hints for Selected Problems
Write the two integers as
Euclidean Algorithm
The greatest common divisor of two integers can be found by listing divisors, but this method becomes inefficient for large numbers. For example, finding
Euclidean Algorithm Revisited
The Euclidean algorithm is one of the oldest and most important algorithms in mathematics. It computes the greatest common divisor of two integers using repeated division.
Functional Equation
The defining series of the zeta function,
Global Class Field Theory
One of the central goals of algebraic number theory is to classify field extensions of a number field
Lattice Reduction
A lattice is a discrete additive subgroup of Euclidean space. More concretely, let
Suggested Projects and Explorations
Study empirical properties of prime numbers through computation.
Least Common Multiples
Let $a$ and $b$ be nonzero integers. An integer $m$ is called a common multiple of $a$ and $b$ if
Computational Aspects
Quadratic residue theory is not only a theoretical subject. It also plays a major role in computational number theory, cryptography, primality testing, and algorithm design.
Analytic Continuation
The defining series of the Riemann zeta function is
Local Class Field Theory
Global class field theory studies finite abelian extensions of number fields such as
Integer Factorization
Integer factorization asks for the prime decomposition of a positive integer. Given
Problem Sets
1. Prove that the sum of two even integers is even.
Greatest Common Divisors
Let $a$ and $b$ be integers, not both zero. An integer $d$ is called a common divisor of $a$ and $b$ if
Higher Reciprocity Laws
Quadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers.
Euler Product Formula
The defining series of the Riemann zeta function is
Hilbert Class Fields
One of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers.
Primality Testing
A prime number is an integer greater than $1$ whose only positive divisors are
The Division Algorithm
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.
Gauss Sums
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.
Definition of the Zeta Function
One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into...
Reciprocity Maps
One of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime.
Fast Integer Arithmetic
Modern computational number theory depends fundamentally on efficient arithmetic with large integers.
Composite Numbers
A positive integer $n>1$ is called composite if it is not prime.
Quadratic Reciprocity
The theory of quadratic residues asks a fundamental question:
Twin Prime Heuristics
A pair of primes
Abelian Extensions
A central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers
Trace Formula
One of the central ideas of modern analysis is that functions may be decomposed spectrally into elementary pieces.
Prime Numbers
Prime numbers are the fundamental building blocks of arithmetic.
Euler Criterion
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 $p$....
Prime Gaps
Let
Adeles and Ideles
The rational numbers may be studied through their completions:
Functoriality
Functoriality is the unifying mechanism of the Langlands program. It predicts systematic relationships between automorphic representations attached to different algebraic groups.
Divisibility Relations
Division of integers does not always produce an integer. For example,
Jacobi Symbol
The Legendre symbol
Short Intervals
The Prime Number Theorem describes the average distribution of primes up to a large number $x$:
Local-Global Principles
A central problem in number theory is determining whether an equation possesses rational or integral solutions.
Langlands Program
The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that...
Glossary
A group $G$ is abelian if
Historical Development of Number Systems
The idea of number arose long before formal mathematics. Early civilizations used numbers for counting objects, measuring land, recording trade, and tracking time.
Legendre Symbol
Let $p$ be an odd prime and let $a\in\mathbb{Z}$. The Legendre symbol is defined by
Error Terms
The Prime Number Theorem states that
Hensel’s Lemma
One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.
Adelic Methods
Number theory studies arithmetic simultaneously at two levels:
Appendix J. Historical Notes and Bibliography
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....
Growth of Integers
The integers extend infinitely in both directions:
Squares Modulo $n$
A quadratic congruence is a congruence involving a square. The basic form is
Logarithmic Integral
The logarithmic integral is the function
Completion of Fields
The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.
Automorphic Representations
Classically, number theory studied special analytic functions such as modular forms. These functions satisfy strong symmetry conditions under actions of arithmetic groups.
Appendix I. Computational Tools
Computation has become an essential part of number theory. Classical arithmetic relied mainly on symbolic reasoning and hand calculations. Modern arithmetic combines rigorous...
Recursive Definitions
Many mathematical objects are defined recursively. A recursive definition specifies:
Geometry of Diophantine Problems
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.
Prime Number Theorem
The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that
Appendix H. Category Theory Basics
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...
$p$-Adic Numbers
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...
Representation Theory Background
Representation theory studies abstract algebraic objects by expressing them as linear transformations of vector spaces.
Strong Induction
Ordinary induction proves a statement $Pn$ by showing that truth passes from one case to the next:
Rational and Integral Points
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:
Chebyshev Bounds
The prime counting function
Appendix G. Linear Algebra Review
A vector space over a field $F$ is a set $V$ equipped with addition and scalar multiplication satisfying the usual algebraic rules.
Absolute Values
The ordinary absolute value on the real numbers measures magnitude:
Weil Conjectures
One of the central problems in arithmetic geometry is understanding the number of solutions of polynomial equations over finite fields.
Mathematical Induction
Many statements in number theory concern all natural numbers. For example, one may wish to prove that
Exponential Diophantine Equations
An exponential Diophantine equation is a Diophantine equation in which one or more unknowns appear as exponents. Typical examples include
Prime Counting Function
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,...
Appendix F. Measure and Integration
Measure theory extends the ideas of length, area, volume, and integration to more general settings. In number theory, measure appears in probability, harmonic analysis,...
Ramification
One of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field.
Étale Cohomology
Classical topology studies geometric spaces using invariants such as homology and cohomology. Over the complex numbers, algebraic varieties can often be viewed as topological...
Absolute Value and Distance
The order relation distinguishes positive and negative integers, but in many situations the sign of a number is less important than its magnitude. For example, the integers
Catalan-Type Equations
A Catalan-type equation is a Diophantine equation involving powers whose values differ by a small amount. The classical example is
Abel Summation
In analytic number theory, one often studies sums of the form
Appendix E. Topology Background
Topology studies continuity, convergence, connectedness, and geometric structure in an abstract setting. In number theory, topology appears naturally in real analysis, complex...
Cyclotomic Fields
One of the most important classes of number fields arises from the solutions of the equation
Arithmetic Surfaces
Arithmetic geometry often studies families of algebraic curves varying over arithmetic bases. The most important base is
Order Relations
The integers are not merely a collection of numbers equipped with arithmetic operations. They also possess an order structure. Given two integers $a$ and $b$, one can...
Sums of Squares
One of the oldest questions in number theory asks which integers can be written as sums of squares. Typical examples are
Convergence Methods
Analytic number theory studies infinite sums, products, and integrals. Before such expressions can be manipulated safely, one must understand the meaning of convergence.
Appendix D. Real and Complex Analysis Review
The real numbers $\mathbb{R}$ extend the rational numbers $\mathbb{Q}$ by filling gaps such as
Finite Fields
The familiar fields
Curves over Fields
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.
Arithmetic Operations
An arithmetic operation is a rule that combines numbers to produce another number. The most basic operations on integers are addition, subtraction, multiplication, and division.
Pell Equations
A Pell equation is a Diophantine equation of the form
Euler Products
Euler products arise when an infinite series has coefficients controlled by multiplication. The simplest and most important example is the zeta series
Appendix C. Abstract Algebra Review
Abstract algebra studies sets equipped with operations. In number theory, these structures organize arithmetic behavior.
Galois Groups
A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider
Morphisms and Fibers
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.
The Integers
The natural numbers are sufficient for counting and addition, but they are not sufficient for subtraction. For example,
Appendix B. Proof Techniques
A mathematical proof is a logically complete argument establishing the truth of a statement from accepted assumptions, definitions, and previously proved results.
Pythagorean Triples
A Pythagorean triple is a triple of positive integers
Infinite Products
An infinite product has the form
Splitting Fields
A central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial
Schemes
Classical algebraic geometry studies varieties defined by polynomial equations. This theory works well over algebraically closed fields, especially over $\mathbb{C}$. However,...
Chapter 1. Foundations of Arithmetic
The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers:
Chapter 2. Classical Number Theory
A Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless...
Chapter 3. Analytic Number Theory
The harmonic series is the infinite series
Chapter 4. Algebraic Number Theory
A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...
Chapter 5. Arithmetic Geometry and Modern Directions
Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations....
Appendix
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 $x \notin A$.
Modern Number Theory
Modern Number Theory book notes exported from ChatGPT, organized into 5 chapters.
Appendix
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 $x \notin A$.
Future Directions in Number Theory
Modern number theory continues to evolve rapidly.
Chronology of Number Theory
| Period | Development |
Index of Definitions
| Definition | Location |
Index of Theorems
| Theorem | Location |
Notation Index
| Symbol | Meaning |
Chapter 5. Arithmetic Geometry and Modern Directions
Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations....
Chapter 4. Algebraic Number Theory
A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...
Chapter 3. Analytic Number Theory
The harmonic series is the infinite series
Chapter 2. Classical Number Theory
A Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless...
Chapter 1. Foundations of Arithmetic
The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers:
LeetCode 952: Largest Component Size by Common Factor
A clear explanation of solving Largest Component Size by Common Factor using prime factorization and union find.
LeetCode 829: Consecutive Numbers Sum
A clear explanation of the Consecutive Numbers Sum problem using arithmetic series formulas and divisibility analysis.
LeetCode 914: X of a Kind in a Deck of Cards
A clear explanation of checking whether card counts share a common group size using the greatest common divisor.
LeetCode 878: Nth Magical Number
A clear explanation of finding the nth magical number using binary search, greatest common divisor, least common multiple, and inclusion-exclusion.
LeetCode 866: Prime Palindrome
A clear explanation of finding the smallest prime palindrome greater than or equal to n by generating odd-length palindromes and testing primality.
LeetCode 858: Mirror Reflection
A clear explanation of Mirror Reflection using room unfolding, least common multiples, and parity.
LeetCode 793: Preimage Size of Factorial Zeroes Function
A clear explanation of finding how many integers have exactly k trailing zeroes in their factorial.
LeetCode 780: Reaching Points
A clear explanation of checking whether one point can reach another by working backward with modulo.
LeetCode 633: Sum of Square Numbers
A two-pointer and number theory solution for checking whether an integer can be written as the sum of two square numbers.
LeetCode 507: Perfect Number
A clear explanation of checking whether a number equals the sum of its positive divisors excluding itself.
LeetCode 365: Water and Jug Problem
A clear explanation of solving the Water and Jug Problem using Bézout's identity and greatest common divisor.
LeetCode 326: Power of Three
A clear explanation of the Power of Three problem using repeated division and integer arithmetic.
LeetCode 263: Ugly Number
A clear explanation of the Ugly Number problem using repeated division by the only allowed prime factors.
LeetCode 258: Add Digits
A clear explanation of the Add Digits problem using repeated digit sums first, then the digital root formula.
11. Number Theory
Integers, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods.