Number theory, also known as higher mathematics, is a branch of mathematics that studies the properties of integers and rational numbers. It is considered one of the oldest branches of mathematics, with a history that dates back to ancient times.
Basic Concepts
- Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.
- Composite Numbers: A composite number is a positive integer that has at least one positive divisor other than one or itself. For example, 4, 6, 8, and 9 are composite numbers.
- Fermat's Little Theorem: If ( p ) is a prime number and ( a ) is an integer not divisible by ( p ), then ( a^{p-1} \equiv 1 \mod p ).
Interesting Theorems
- Euler's Theorem: For any integer ( a ) and any integer ( n ) that is coprime to ( a ), ( a^{\phi(n)} \equiv 1 \mod n ), where ( \phi(n) ) is Euler's totient function.
- Fermat's Last Theorem: It states that no three positive integers ( a ), ( b ), and ( c ) can satisfy the equation ( a^n + b^n = c^n ) for any integer value of ( n ) greater than 2.
Further Reading
To learn more about number theory, you can explore the following resources:
Prime Numbers
Fermat's Little Theorem
Euler's Theorem
Fermat's Last Theorem