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An upper bound in Goldbach’s problem. (English) Zbl 0780.11047

There are many partial results on Goldbach’s conjecture that every even \(n>2\) is a sum of two primes: for example, every even \(n\) is a sum of 6 primes (Ramaré); every sufficiently large even \(n\) is a sum of a prime and a number with at most two prime factors (Chen); at most \(c\cdot x^{1-\delta}\) even \(n\leq x\) cannot be written as the sum of two primes (Montgomery and Vaughan).
This paper considers the maximal number of ways an even \(n\) can be written as the sum of two primes. Clearly the optimal case is if for every prime \(q\) with \(n/2\leq q<n-2\), then \(p=n-q\) is also prime. The authors show that \(n=210\) is the last number for which this occurs. This curiosity is proved by first ruling out, on theoretical grounds, all \(n\geq 2\cdot 10^{24}\), and then using a computation to rule out all \(n\) with \(210<n<2\cdot 10^{24}\). The main tool is a combinatorial lemma which permits ruling out all \(n\) in intervals roughly of size \([3x/2,2x]\). Effective sieve estimates, together with the Rosser- Schoenfeld prime number theorem, lead to the bound \(2\cdot 10^{24}\). Numerical implementation of the lemma, using a rapid primality test due to Proth, deals with the finite interval.
Reviewer: R.Rumely (Georgia)

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11Y35 Analytic computations
11N36 Applications of sieve methods
11Y11 Primality
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