科研之友
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摘要
When K %26gt;= 1 is an integer and S is a set of prime numbers in the interval (N/2, N] with |S| %26gt;= pi*(N)/K, where pi*(N) is the number of primes in this interval, we obtain an upper bound for the additive energy of S, which is the number of quadruples (x (1), x (2), x (3), x (4)) in S-4 satisfying x(1)+x(2) = x(3)+x(4). We obtain this bound by a variant of a method of Ramar, and I. Ruzsa. Taken together with an argument due to N. Hegyvari and F. Hennecart, this bound implies that when the sequence of prime numbers is coloured with K colours, every sufficiently large integer can be written as a sum of no more than CK log log 4K prime numbers, all of the same colour, where C is an absolute constant. This assertion is optimal up to the value of C and answers a question of A. Sarkozy.
- 出版日期2014-3
