Is every even number greater than 4 the sum of two odd primes? Are there infinitely many primes *p* such that *p* + 2 is also a prime? Is every sufficiently large positive integer the sum of four cubes? These three questions are all famous unsolved problems in number theory: the first is called the *Goldbach conjecture*, the second is the *twin prime conjecture* (discussed in some detail in ANALYTIC NUMBER THEORY [IV.2]), and the third is a special case of *Waring’s problem*, which we shall discuss later.

These three problems belong to an area of mathematics known as *additive number theory*. In order to say in general terms what this area is, it is useful to make some simple definitions. Suppose that ...

Start Free Trial

No credit card required