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We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p 13 + ... + p j 3 with |p i − (N/j)1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ ij), for δ j = 1/45; 1/30; 1/25; 2/45, respectively.  相似文献   

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We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 +p2^3 +... +p9^3 with [pj - (N/9)^1/31 ≤ N^(1/3)-θ, where pj, j = 1,2,...,9, are primes and θ = (1/51) -ε.  相似文献   

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本文证明了对5≤s≤8,几乎所有的满足某些同余条件的正整数N都可以表示为N=p31+···+p3s,|pi-(N/s)1/3|≤N1/3-θs,其中θ5=7261-2ε,θ6=5159-ε,θ7=11333-ε,θ8=19561-ε.  相似文献   

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The well-known Goldback-Vinogradov theorem states that every large odd integer is a sum of three primes. In the present paper it is further proved that every large odd integerN can be represented as
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The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.  相似文献   

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It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. Using the circle method and sieves, we prove that the conjecture is true for at least 1.5% of the positive integers satisfying the necessary conditions.  相似文献   

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In 1965 Erd?s introduced f2(s): f2(s) is the smallest integer such that every l>f2(s) is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, f2(s)?p2+p3+?+ps+1+3106, and the set of s with the equality has the density 1.  相似文献   

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