Rational Number Theory in the 20th Century

The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat's problem. "Rational Number Theory in the 20th Century: From PNT to FLT" offers a short survey of 20th century developments in classical number theory, documenting betwee

An upper bound in Goldbach's problem

: It is clear that the number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval [n=2; n \Gamma 2]. We show that 210 is the largest value of n for which this upper bound is attained. 1. Introduction. In 1742 Christian Goldbach wrote, in a letter to Euler, that on the evidence of extensive computations he was convinced that every integer exceeding 6 was the sum of three primes. Euler replied that if an even number 2n + 2 is so represented then one of those primes must be even and thus 2, so that every even number 2n, greater than 2, can be represented as the sum of two primes; it is easy to see that this conjecture implies Goldbach's original proposal, and it has widely become known as Goldbach's conjecture. Although still unresolved, Goldbach's conjecture is widely believed to be true. It has now been verified for every even integer up to 2 \Theta 10 10 (in [3]), and there are many interesting partial results worthy ..