Number Theory — Definition, Formula & Examples
Number theory is the branch of mathematics that studies the properties of integers — especially prime numbers, divisibility, and the relationships between whole numbers.
Number theory is the mathematical discipline concerned with the structure and behavior of the integers , including the study of prime factorization, congruences, Diophantine equations, and arithmetic functions.
How It Works
Number theory asks questions about whole numbers and looks for patterns or proofs. For example: Which numbers are prime? Can every even number greater than 2 be written as a sum of two primes? How many primes are there below a given number? You use tools like divisibility rules, the Euclidean algorithm, modular arithmetic, and prime factorization to answer these questions. Many results in number theory are simple to state but surprisingly deep to prove.
Worked Example
Problem: Use the Euclidean algorithm (a classic number theory method) to find the greatest common divisor of 84 and 30.
Step 1: Divide 84 by 30 and find the remainder.
Step 2: Replace 84 with 30 and 30 with the remainder 24. Divide again.
Step 3: Replace 30 with 24 and 24 with 6. Divide again.
Answer: The remainder is 0, so the GCD is the last nonzero remainder: .
Why It Matters
Number theory is the mathematical backbone of modern cryptography. RSA encryption, which secures online banking and private communications, relies directly on the difficulty of factoring large numbers into primes. The field also appears in competition math, computer science algorithms, and coding theory.
Common Mistakes
Mistake: Thinking number theory only covers basic arithmetic like addition and multiplication.
Correction: Number theory goes far beyond computation. It investigates deep structural questions about integers, such as the distribution of primes, solutions to equations in integers, and properties of congruences.