Number theory
Branch · 3rd century BC onward · Mathematics, stage 2: Map the branches
Number theory studies the whole numbers: divisibility, primes, factorisation, and the equations that whole numbers do or do not satisfy. It begins in the Elements, where Euclid proves that the primes are infinite in number and gives an algorithm for greatest common divisors, and it becomes a systematic subject with Gauss in 1801. Its questions are famous for being easy to state and extremely hard to settle, and for staying useless until suddenly they are not: the difficulty of factoring large numbers now underwrites internet cryptography.
What it claims
- The whole numbers have structure worth studying for its own sake, independently of any application.
- Primes are the multiplicative building blocks of the whole numbers, and each number factors into them in only one way.
- A statement about numbers can be checked in billions of cases and still be unproved, so computation is evidence and not proof.
Key ideas
People
Sources
- Number theory Wikipedia
- Prime numbers MacTutor History of Mathematics
