PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Number Theory ↓


Elementary Number Theory


Elementary number theory is a branch of mathematics that deals with the properties and relations of numbers, particularly integers. It is called "elementary" because it deals with basic concepts that are foundational to further study in mathematics. Historically, number theory focused on the study of integers and their properties, such as divisibility and primality. The field remains a cornerstone of modern mathematics, captivating mathematicians with its beauty, simplicity, and depth.

1. Basics of numbers

To understand number theory, we must first understand what numbers are. Numbers can be considered basic abstract concepts that we use to count, measure, and label. In elementary number theory, we deal mainly with integers, which are whole numbers that can be positive, negative, or zero.

Let's take a look at some of the basic types of numbers that we often encounter:

  • Natural numbers: These are the numbers by which we count naturally like 1, 2, 3, etc.
  • Whole numbers: These are the natural numbers including zero, such as 0, 1, 2, 3, etc.
  • Integers: These include whole numbers and their negative counterparts, such as -2, -1, 0, 1, 2, etc.

2. Divisibility

Divisibility is a fundamental concept in number theory. An integer a is divisible by another integer b if there exists an integer c such that a = b times c. If a is divisible by b, we say that b divides a, or b is a divisor of a.

Example:

Consider a = 15 and b = 3.

15 = 3 times 5

So 3 divides 15.

3. Prime numbers

Prime numbers are the building blocks of integers. A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. In other words, it cannot be formed by multiplying two smaller natural numbers.

Examples of prime numbers:

  • 2
  • 3
  • 5
  • 7
  • 11

Each of these numbers cannot be evenly divided by any number other than 1 and itself.

4. Common divisors and multiples

Two important concepts in number theory are common divisors and common multiples.

  • Greatest Common Divisor (GCD): The largest number that can divide two or more integers without leaving a remainder.
  • Least Common Multiple (LCM): The smallest number that is a multiple of two or more integers.

Example:

Find the GCD and LCM of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Common divisors: 1, 2, 3, 6

So, GCD(12, 18) = 6.

Multiples of 12: 12, 24, 36, 48, ...

Multiples of 18: 18, 36, 54, 72, ...

Common multiples: 36, 72, ...

Hence, LCM(12, 18) = 36.

5. Congruence

Congruence is a way of expressing that two numbers leave the same remainder when divided by another number. If two numbers leave the same remainder when divided by a number, they are said to be congruent with respect to that number. This can be written as:

a equiv b  (text{mod}  n)

where a and b are identical in modulus of n.

Example:

24 is equivalent to 9 in the ratio of 5, since both have a remainder of 4.

24 equiv 9  (text{mod}  5)

6. Diophantine equations

Diophantine equations refer to polynomial equations where the solutions are restricted to integers. They are named after the ancient Greek mathematician Diophantus. A simple example is the linear Diophantine equation:

ax + by = c

where x and y are integers. If the GCD of a and b divides c then the solution exists.

Example:

Solve 3x + 6y = 18.

The GCD of 3 and 6 is 3, and 3 divides 18, so a solution exists.

A simple solution is x = 0 and y = 3.

7. Fundamentals of arithmetic

The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers, up to order. This is the basis for many other concepts in number theory.

Example:

Consider the number 60.

60 = 2^2 times 3 times 5

This representation is unique (regardless of the order).

8. Whole numbers

A perfect number is an integer that is the sum of its proper positive divisors, not including itself. The smallest perfect number is 6.

Example:

Consider the number 6.

Divisors of 6: 1, 2, 3

The sum of these divisors is 1 + 2 + 3 = 6.

9. Euler's totient function

Euler's totient function, denoted as phi(n), is a function that returns the number of integers up to n that are coprime with n. Two numbers are coprime if their greatest common divisor is 1.

Example:

Calculate phi(9).

Coprime numbers of 9 are 1, 2, 4, 5, 7, 8.

So, phi(9) = 6.

10. Conclusion

Elementary number theory is the backbone of our understanding of mathematics as it deals with the properties and relationships of integers. Through its basic concepts, such as divisibility, prime numbers, and congruence, we gain insight into more complex and abstract parts of mathematics. These foundational topics not only develop critical thinking and problem-solving skills, but also have applications in cryptography, computer science, and other fields.

As we have seen, this subject is full of interesting problems and theorems, many of which have not only stood the test of time but continue to inspire new generations of mathematicians. While this overview covers the basics of elementary number theory, much awaits exploration in more advanced study, providing a deeper understanding of the beautiful world of integers.


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Elementary Number Theory

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