In 2002, Heath-Brown and Puchta  applied a rather different approach to this problem and showed that k = 13 and, on the GRH, k = 7. In 2003, Pintz and Ruzsa  established this latter result and announced that k = 8 is acceptable unconditionally. This paper is yet to appear in print. Elsholtz, in an unpublished manuscript, showed that k = 12; this was proved independently by Liu and Lü .
They showed that k = 161 is acceptable and Platt and Trudgian  revised it to 156.
where k is a positive integer. She proved that the simultaneous equations (1.5) are solvable for k = 63. Then Platt and Trudgian  revised it to 62.
where k is a positive integer. Our result is stated as follows.
For k = 455, the equations (1.6) are solvable for every pair of sufficiently large positive even integers N1 and N2 satisfying N2≫ N1 > N2.
Throughout this paper, the letter ϵ denotes a positive constant which is arbitrarily small but may not be the same at different occurrences. And p and v denote a prime number and a positive integer, respectively.
2 Outline of the method
Here we give an outline for the proof of Theorem 1.1.
It follows from 2Pi≤ Qi that the arcs 𝓜1(a1, q1) and 𝓜2(a2, q2) are mutually disjoint respectively.
where i = 1,2, e(x): = exp(2 πix) and L = log2 N1.
where Rst(N1, N2) denotes the combination of s-th term in the first bracket and the t-th term in the second bracket.
We will establish Theorem 1.1 by estimating the term Rst(N1, N2) for all 1≤ s, t ≤3. We need to show that R(N1, N2) > 0 for every pair of sufficiently large odd positive integers N2≫ N1 > N2.
We need the following lemmas to prove Theorem 1.1.
with E(0.9457) > 109/126+10−10.
This is Lemma 4.4 in Liu and Lü .
For all integers n≡ 0 (mod 2), we have 𝔖(n)≫ 0.2448.
Then we get the proof of this lemma. □
The proof of this lemma can be found in , which is based on the estimate of exponential sums over primes. □
Thus we can get the proof of this lemma. □
3 Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1.
where we used
and get k≥ 455. Consequently, we deduce that every pair of large odd integers N1, N2 satisfying N2≫ N1 > N2 and N1≡ N2 ≡ 0(mod 2) can be written in the form of (1.3) for k≥ 455. Thus Theorem 1.1 follows.
This work is supported by Natural Science Foundation of China (Grant Nos. 11761048). The authors would like to express their thanks to the referee for many useful suggestions and comments on the manuscript.
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