Basic Concepts in Elementary Number Theory


1. Basic Concepts in Number Theory

a) Integers:

  • The set of integers (Z\mathbb{Z}) includes positive and negative whole numbers as well as zero: Z={,3,2,1,0,1,2,3,}\mathbb{Z} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \dots \}

b) Divisibility:

  • An integer aa divides another integer bb (denoted as aba \mid b) if there exists an integer kk such that b=akb = ak

  •  For example, 393 \mid 9 because 9=3×39 = 3 \times 3

c) Prime Numbers:

  • A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. For example, 2,3,5,7,11,132, 3, 5, 7, 11, 13 are prime numbers.

d) Greatest Common Divisor (GCD):

  • The GCD of two integers aa and bb is the largest number that divides both aa and bb.
  •  For example, gcd(12,8)=4

e) Least Common Multiple (LCM):

  • The least common multiple of two integers aa and bb is the smallest positive integer divisible by both. For example, lcm(12,8)=24\text{lcm}(12, 8) = 24.

2. Prime Numbers and Factorization

a) Fundamental Theorem of Arithmetic:

  • Every integer greater than 1 can be factored uniquely into a product of prime numbers. This is the basis of prime factorization. For example: 60=22×3×560 = 2^2 \times 3 \times 5

b) Sieve of Eratosthenes:

  • A method to find all prime numbers less than a given nn. By eliminating multiples of each prime starting from 2, we identify the primes in a range.

3. Modular Arithmetic

a) Concept:

  • Modular arithmetic involves numbers wrapping around after reaching a certain value, called the modulus. For example, 120 (mod 12)12 \equiv 0 \ (\text{mod} \ 12) since 12 leaves a remainder of 0 when divided by 12.

b) Basic Properties:

  • Modular arithmetic obeys the basic rules of addition, subtraction, and multiplication. For example: (7+9)mod5=16mod5=1(7 + 9) \mod 5 = 16 \mod 5 = 1

4. Linear Congruences

a) Definition:

  • A linear congruence takes the form axb (mod n)ax \equiv b \ (\text{mod} \ n)where a,b,a, b, and nn are integers. Solving for xx involves finding values of xx that satisfy the equation within the modular system.

b) Solving Linear Congruences:

  • Solving these equations often requires using the Euclidean algorithm to find the multiplicative inverse of aa modulo nn.

5. Diophantine Equations

a) Definition:

  • Diophantine equations are polynomial equations where integer solutions are sought. For example: ax+by=cax + by = c where a,b,ca, b, c are integers and x,yx, y are unknowns.

b) Solutions:

  • A solution to this equation exists if and only if gcd(a,b)\text{gcd}(a, b) divides cc.

6. Theorems in Number Theory

a) Fermat's Little Theorem:

  • If pp is a prime and aa is not divisible by pp, then: ap11 (mod p)
  •  This theorem has applications in cryptography and computational mathematics.

b) Euler’s Theorem:

  • For any integer aa coprime to nn: aÏ•(n)1 (mod n)a^{\phi(n)} \equiv 1 \ (\text{mod} \ n)where Ï•(n)\phi(n) is Euler’s totient function, counting the integers up to nn that are coprime to nn.

7. Cryptography and Number Theory

a) RSA Encryption:

  • RSA encryption is a public-key cryptographic system based on the difficulty of factoring large integers. It relies on modular arithmetic and the properties of prime numbers.

b) Diffie-Hellman Key Exchange:

  • This method uses modular exponentiation to enable two parties to securely exchange cryptographic keys.

8. Applications of Elementary Number Theory

a) Cryptography:

  • Elementary number theory is central to encryption methods, such as RSA, which secure online communications.

b) Computer Algorithms:

  • Many algorithms, like those for primality testing, are rooted in number theory. Modular arithmetic is also essential for hash functions in computer science.

c) Pure Mathematics:

  • Number theory provides tools for exploring deep mathematical problems, including unsolved questions like the Goldbach conjecture or the Riemann hypothesis.



Some Examples

1. Divisibility and Prime Numbers

Example 1: Divisibility

  • Determine whether 185418 \mid 54 (i.e., check if 18 divides 54).
    • Solution: 54÷18=354 \div 18 = 3 which is an integer. Therefore, 185418 \mid 54.

Example 2: Prime Numbers

  • Check if 29 is a prime number.
    • Solution: To verify, we check divisibility by all prime numbers less than 29\sqrt{29} (i.e., 2, 3, 5). Since 29 is not divisible by any of them, 29 is prime.

2. Greatest Common Divisor (GCD)

Example 3: Using Euclidean Algorithm

  • Find the GCD of 56 and 98 using the Euclidean algorithm.
    • Solution: Apply the division algorithm: 98=1×56+4298 = 1 \times 56 + 4256=1×42+1456 = 1 \times 42 + 1442=3×14+042 = 3 \times 14 + 0 Since the remainder is 0, the GCD is the last non-zero remainder, gcd(56,98)=14\text{gcd}(56, 98) = 14.

3. Modular Arithmetic

Example 4: Modular Addition

  • Compute (7+10)mod6
    • Solution: First, add the numbers: 7+10=177 + 10 = 17 Now, divide by 6 and take the remainder: 17÷6=2 remainder 517 \div 6 = 2 \text{ remainder } 5Therefore, 175 (mod 6)17 \equiv 5 \ (\text{mod} \ 6).

Example 5: Modular Multiplication

  • Compute (5×9)mod7.
    • Solution: First, multiply the numbers: 5×9=455 \times 9 = 45 Now, divide by 7 and take the remainder: 45÷7=6 remainder 345 \div 7 = 6 \text{ remainder } 3Therefore, 453 (mod 7)45 \equiv 3 \ (\text{mod} \ 7)

4. Linear Diophantine Equation

Example 6: Solving a Diophantine Equation

  • Solve 15x+21y=315x + 21y = 3 for integer values of xx and yy.
    • Solution: First, find the GCD of 15 and 21, which is 3 (since 3153 \mid 15 and 3213 \mid 21).
    • Divide the equation by 3: 5x+7y=15x + 7y = 1 Using the extended Euclidean algorithm, we solve for xx and yy: 7=1×5+27 = 1 \times 5 + 25=2×2+15 = 2 \times 2 + 1Working backwards: 1=52×2=52×(71×5)=3×52×71 = 5 - 2 \times 2 = 5 - 2 \times (7 - 1 \times 5) = 3 \times 5 - 2 \times 7 Thus, x=3y=2y = -2 is a particular solution to the equation.

5. Fermat's Little Theorem

Example 7: Using Fermat’s Little Theorem

  • Find 2100mod132^{100} \mod 13.
    • Solution: Fermat's Little Theorem states that for a prime pp, if aa is not divisible by pp then: ap11 (mod p)a^{p-1} \equiv 1 \ (\text{mod} \ p) Here, p=13p = 13 and a=2 so:2121 (mod 13)2^{12} \equiv 1 \ (\text{mod} \ 13)Now, write 100=8×12+4 so:2100=(212)8×2418×24 (mod 13)2^{100} = (2^{12})^8 \times 2^4 \equiv 1^8 \times 2^4 \ (\text{mod} \ 13) Compute 24=162^4 = 16 and 16mod13=3 Therefore:21003 (mod 13)2^{100} \equiv 3 \ (\text{mod} \ 13)

These examples illustrate key concepts of divisibility, modular arithmetic, GCD, Diophantine equations, and theorems like Fermat’s Little Theorem.

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