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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.27 Problems and Results in Additive Number Theory

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 ...

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