Small prime solutions to diagonal Diophantine equations

摘要

Let k = 3 be an integer, s = 2(k) + 1, (a(i), a(j)) = 1, 1 = i j = s, where a(1), ..., a(s) are nonzero integers, and let n be an integer. Suppose that a(1), ..., a(s) satisfy some necessary congruent conditions. In this paper, it is proved that(i) if a(j) are not all of the same sign, then the equation a(1)p(1)(k) + ... + a(s)p(s)(k) = n has prime solutions satisfying p(j) vertical bar n vertical bar(1/k) + max {vertical bar a(j)vertical bar}(C(k)+epsilon),(ii) if all a(j) are positive and n max{vertical bar a(j)vertical bar}(kC(k)+1+epsilon), then a(1)p(1)(k) + ... + a(s)p(s)(k) = n is solvable in primes p(j), @where C(k) = 1 + 6.2(k) + 20/3.2(2k-1) + 3.2(k-1-20)Our result uses C(3) = 39/22 approximate to 1.7727 ... as an improvement of the recent result C(3) = 2 due to Zhao (2016), and largely improves the results C(k) = 3.2(k-1) for k = 4 proved by Yang and Hu (2016).