ABSTRACT
The development of number theory has always been driven by problems. People play with numbers and find patterns. Do these patterns persist? Can we prove them? In this chapter, we discuss some of the more prominent examples that have influenced number theory over the years.
In 1742, Christian Goldbach wrote a letter to Euler conjecturing that every even integer from 4 onward can be written as a sum of two primes. For example, 100 = 29 + 61 and 32 = 13 + 19. This has become known as Goldbach’s Conjecture. No one had any idea how to prove it, and there was really no progress until 1920, when Viggo Brun showed that every even integer is a sum of two integers, each of which has at most 9 prime factors. This led to the development of sieve theory, which is a vast generalization of the Sieve of Eratosthenes. The techniques were continually refined, and in 1966, Jingrun Chen showed that every sufficiently large even integer is a sum of a prime and an integer that is either prime or the product of two primes. The related question for odd integers was studied in the 1930s by Vinogradov, who showed that every sufficiently large odd integer is a sum of three primes. The problem was completed in 2013 by Helfgott, who showed that every primes.
