On a Diophantine equation involving fractional powers with primes of special types

1.   Introduction

For a fixed integer $k\geq1$ and sufficiently large integer $N$, the well-known Waring-Goldbach problem is devoted to investigating the solvability of the following Diophantine equality

in prime variables $p_1, p_2, \ldots, p_s$. Numerous mathematicians have derived many splendid results in this field. For instance, in 1937, Vinogradov ^[25] proved that such a representation of the type (1.1) exists for every sufficiently large odd integer $N$ with $k = 1, s = 3$. Later in 1938, based upon Vinogradov's work, Hua ^[9] showed that (1.1) is solvable for every sufficiently large integer $N$ satisfying that $N\equiv5\; (\text{mod}\; 24)$ with $k = 2, s = 5$.

In 1933, Segal ^[21,22] studied the following anolog of the well-known Waring problem. Suppose that $c > 1$ and $c\notin \mathbb{N}$; there exists a positive integer $s = s(c)$ such that for every sufficiently large natural number $N$, the equation

has a solution with $m_{1}, m_{2}, \ldots, m_{s}$ integers, where $[t]$ denotes the integral part of any $t\in \mathbb{R}$.

To obtain a result that is analogous to the ternary Goldbach problem, in 1995, Laporta and Tolev ^[13] considered the equation

where $p_{1}, p_{2}, p_{3}$ are prime numbers, $c\in\mathbb{R}, c > 1, N\in\mathbb{N},$ and $[t]$ denotes the integral part of $t$. They proved that if $1 < c < \frac{17}{16}$ and $N$ is a sufficiently large integer, then the Eq (1.2) has a solution in prime numbers $p_{1}, p_{2}, p_{3}$. Later, the upper bound of $c$ was enlarged to

by Kumchev and Nedeva ^[12], Zhai and Cao ^[27], Cai ^[6], Li and Zhang ^[15], and Baker ^[2], successively and respectively.

On the other hand, as an analogue of Hua's theorem on five prime squares, Li and Zhang ^[14] first studied the solvability of the Diophantine equation

in prime numbers $p_1, p_2, p_3, p_4, p_5.$ They proved that if $1 < c < \frac{4109054}{1999527}, c\neq2$ and $N$ is a sufficiently large integer, then the Eq (1.3) has a solution in prime numbers $p_{1}, p_{2}, p_{3}, p_4, p_5$. Later this result was improved by Li ^[17] who enlarged the upper bound for $c$ to $\frac{408}{197},$ and by Baker ^[2] who replaced $\frac{408}{197}$ by $\frac{609}{293}$.

For any natural number $r$, let $\mathscr{P}_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. There are many papers that are devoted to the study of problems involving primes of a special type. In 1973, Chen ^[4] established that there exist infinitely many primes $p$ such that $p+2$ has at most 2 prime factors. In 2000, Tolev ^[24] proved that for every sufficiently large integer $N\equiv3 \; (\text{mod}\; 6)$, the equation

has a solution in prime numbers $p_1, p_2, p_3$ such that $p_1+2\in\mathscr{P}_{2}, p_2+2\in\mathscr{P}_{5}, p_3+2\in\mathscr{P}_{7}$. After that, this result was improved by some mathematicians, and the best result in this field was obtained by Matomäki and Shao ^[18], who showed that for every sufficiently large integer $N\equiv3\; (\text{mod}\; 6)$ the Eq (1.4) has a solution in prime numbers $p_1, p_2, p_3$ such that $p_1+2, p_2+2, p_3+2\in\mathscr{P}_{2}.$

Bearing in mind the result of ^[18], it is natural for us to conjecture that if $c$ is close to 1, then the Eq (1.2) is solvable in primes $p_1, p_2, p_3$ such that $p_i+2\in\mathscr{P}_{2}$. An attempt to establish this kind of the result was first made by Petrov ^[19], who showed that, for $1 < c < \frac{17}{16}$ and every sufficiently large integer $N$, the Eq (1.2) is solvable in prime numbers $p_1, p_2, p_3$ such that each of the numbers $p_i+2$ has at most $[\frac{95}{17-16c}]$ prime factors, counted according to multiplicity. Recently, Li et al. ^[16] improved Petrov's result; they extended the range of $c$ to $1 < c < \frac{2173}{1930}$ and reduced the number of prime factors of $p_i+2, i = 1, 2, 3$ to $[\frac{11387}{4346-3860c}].$

Referencing Hua's work, Tolev ^[24] also showed that for every sufficiently large integer $N\equiv5 \; (\text{mod}\; 24)$, the equation

has a solution in prime numbers $p_1, p_2, p_3, p_4, p_5$ such that $p_1+2\in\mathscr{P}_{2}, p_2+2\in\mathscr{P}_{2}, p_3+2\in\mathscr{P}_{5}, p_4+2\in\mathscr{P}_{5}$ and $p_5+2\in\mathscr{P}_{8}$. And later in 2009, Cai and Lu ^[5] improved Tolev's result by showing that the Eq (1.5) has a solution in prime numbers $p_1, p_2, p_3, p_4, p_5$ such that $p_1+2\in\mathscr{P}_{2}, p_2+2\in\mathscr{P}_{2}, p_3+2\in\mathscr{P}_{4}, p_4+2\in\mathscr{P}_{4}$ and $p_5+2\in\mathscr{P}_{5}$. Motivated by Petrov ^[19] and Tolev ^[24], it is reasonable to conjecture that if $N$ is a sufficiently large natural number and $c$ is close to 2, then the Eq (1.3) has a solution in prime numbers $p_1, p_2, p_3, p_4, p_5$ such that $p_i+2$ are almost-primes of a certain fixed order.

In this paper, we shall prove the following result.

Theorem 1.1. Suppose that $2 < c < \frac{990}{479}$ and let $N$ be a sufficiently large natural number. Then the equation (1.3) has a solution in prime numbers $p_1, p_2, p_3, p_4, p_5$ such that each of the numbers $p_1+2, p_2+2, p_3+2, p_4+2$ and $p_5+2$ has at most $\left[\frac{6227}{3960-1916c}\right]$ prime factors, counted with the multiplicity.

2.   Preliminaries

Throughout this paper, the letter $p$, with or without subscript, always stand for prime numbers. We use $\varepsilon$ to denote a sufficiently small positive number, and the value of $\varepsilon$ may change from statement to statement. As usual, we use $\mu(n), \Lambda(n), \varphi(n)$ and $\tau(n)$ to denote Möbius' function, von Mangolds' function, Euler's function and the Dirichlet divisor function, respectively. We write $f = \mathcal{O}(g)$ or, equivalently, $f \ll g$ if $|f|\leq Cg$ for some positive number $C$. If we have simultaneously, that $A\ll B$ and $B\ll A$, then we shall write $A\asymp B$. Moreover, we shall use $(m, n)$ and $[m, n]$ for the greatest common divisor and the least common multiple of the integers $m$ and $n$, respectively. And we use $e(\alpha)$ to denote $e^{2\pi i\alpha}$. In addition, we define

Lemma 2.1. Suppose that $D > 4$ is a real number and let $\lambda^{\pm}(d)$ represent the Rosser functions of level $D$. Then we have the following properties:

(1) For any positive integer d we have

(2) If $n\in\mathbb{N}$ then

(3) If $z\in\mathbb{R}$ and if

then we have

where $F(s)$ and $f(s)$ denote the classical functions in the linear sieve theory that are respectively defined by

and

Here $\gamma$ denotes the Euler constant.

Proof. This is a special case of the work by Greaves ^[8]. □

Lemma 2.2. Let

Then we have

Proof. The proof is the same as in Lemma 13 of ^[3]. □

Lemma 2.3. Suppose that $f(x):[a, b]\rightarrow \mathbb{R}$ has continuous derivatives of arbitrary order on $[a, b]$, where $1\leq a < b\leq2a$. Suppose further that

Then for any exponential pair $(\kappa, \lambda)$, we have

Proof. See (3.3.4) of ^[7]. □

Lemma 2.4. For any complex number $z_{n}$, we have

where $Q$ is any positive integer.

Proof. See Lemma 8.17 of ^[11]. □

Lemma 2.5. Let $t$ be a non-integer, $\alpha\in(0, 1)$ and $H\geq3.$ Then we have

where

Proof. See Lemma 12 of ^[1]. □

Lemma 2.6. For any real number $\theta$, we have

where

Proof. See (3) of ^[10]. □

Lemma 2.7. Let $f(x)$ be a real differentiable function such that $f'(x)$ is monotonic and $f'(x)\geq m > 0,$ or $f'(x)\leq-m < 0,$ throughout the interval $[a, b]$. Then

Proof. See Lemma 4.2 of ^[23]. □

Lemma 2.8. Suppose that $M > 1, c > 1, c\notin \mathbb{Z}$ and $\gamma > 0.$ Let $\mathscr{A}(M; c, \gamma)$ denote the number of solutions of the following inequalities

Then we have

Proof. See Theorem 2 of ^[20]. □

3.   Beginning of the proof

The central focus of this paper is the study of the sum

In order to prove Theorem 1.1, we need only to show that $\Gamma > 0.$ By the trivial orthogonality relation given by

we can write $\Gamma$ as

By the definition of $\Lambda_{i}$ in Lemma 2.2, we can see that

Then by Lemma 2.2 we find that

By the symmetric property, we have

Hence, by (3.1) and (3.2) we obtain

Now define

Consider $\Gamma_1$ first. By (3.4) we can derive that

where

Similarly, we have

Now combining (3.3), (3.5) and (3.8) we get

In the following sections, we shall prove that

4.   The integrals $\Gamma_{11}$ and $\Gamma_{61}$

In this section, we will give an asymptotic formula for the integrals $\Gamma_{11}$ and $\Gamma_{61}$ defined by (3.6) and (3.8), respectively. We consider the sum

where $\lambda(d)$ are real numbers satisfying

Furthermore, we define

Lemma 4.1. Let $L(\alpha)$ and $I(\alpha)$ be defined by (4.1) and (4.3), respectively. Suppose that $\xi$ and $\delta$ satisfy the following conditions

Then for $|\alpha|\leq\tau$, we have

where $A > 0$ is a sufficiently large constant.

Proof. See Lemma 2.8 in ^[16]. □

Lemma 4.2. Let $L(\alpha)$ and $I(\alpha)$ be defined by (4.1) and (4.3), respectively. Then we have

Proof. By using the trivial estimate $|I(\alpha)|\ll X$ and $L(x)\ll X\log^2X,$ (ⅰ) and (ⅱ) follow from Lemma 2.9 in ^[16]. For (ⅲ), we have

where Lemma 2.8 is applied in the last step. □

Let

Then we can easily get the elementary estimate

By using Lemma 4.2 and (4.4) we find that

Let

Then we can derive from Lemma 4.2, (3.6), (4.5) and (4.6) that

Define

Then by the argument used in Lemma 2.13 of ^[16], we have

With the help of Lemma 2.7 we can get that $I(\alpha)\ll |\alpha|^{-1}X^{1-c}$. Hence, from (4.6) and (4.8) we find that

Now combining (4.7) and (4.10) we obtain

Similarly, we can prove that

5.   The upper bound for the integrals $\Gamma_{12}$ and $\Gamma_{62}$

In this section we shall consider the upper bound for the integrals $\Gamma_{12}$ and $\Gamma_{62}$ defined by (3.7) and (3.8), respectively. Define

Lemma 5.1. For $\alpha\in(0, 1)$, we have

Proof. The proof is exactly the same as that of Lemma 2.7 in ^[16], where the exponential pair $(\kappa, \lambda) = (\frac{1}{9}, \frac{13}{18})$ is used. One can see ^[16] for details. □

Lemma 5.2. Let $f(n)$ be a complex valued function defined on $n \in(\frac{X}{2}, X]$. Then we have

where

and where $a(k), c(k)$ are real numbers satisfying

Proof. The proof can be found on page 112 of ^[26]. □

Lemma 5.3. Suppose that $2 < c < \frac{990}{479}.$ Let $\lambda(d)$ be real numbers that satisfy (4.2) and $L(\alpha)$ be defined by (4.1). Then we have

Proof. From (4.1), it is easy to see that

where

Hence by (5.1) we need only to show that the estimation

holds for $2 < c < \frac{990}{479}.$ Let $H = X^{\frac{8}{479}}.$ Then, by using Lemma 2.5, we get

By Lemmas 2.3 and 2.6 with the exponential pair $(\kappa, \lambda) = (\frac{1}{6}, \frac{2}{3})$, we obtain

Next we consider the first term on the right-hand side of (5.3). We write it in the following form

where

By applying Lemma 5.2 we find that

where

and $|a(k)|\leq\tau(k), |c(k)|\leq\log k$. Clearly, by (5.8) we can write $S_{2}$ as

where

Therefore, by (5.6) and (5.10) we have

We consider the sum $S_{21}$ defined by (5.11) first. We change the order of the summation to write it in the following form

Since $\lambda(d) = 0$ for $2|d$, from the condition $d|k\ell+2$ we have that $(k, d) = 1.$ Hence there exists an integer $\ell_{0}$ such that $k\ell+2\equiv0\; (\text{mod}\; d)$ is equivalent to $\ell\equiv\ell_{0}\; (\text{mod}\; d)$, which means that $\ell = \ell_{0}+md$ for some integer $m$. Therefore, we get

By using Lemma 2.3 with the exponential pair $(\kappa, \lambda) = A^2BA^2B(0, 1) = (\frac{1}{20}, \frac{33}{40})$ we find that the sum over $m$ in (5.13) is given by

Then from (4.2), (5.13) and (5.14) we can obtain

For the sum $S_{1}$ given by (5.7), we can apply partial summation to get rid of the $\log$ factor and then proceed as in the same process for $S_{21}$ to get

Now we consider the sum $S_{3}$. By a splitting argument, we can decompose $S_{3}$ into $\mathcal{O}(\log X)$ sums of the following form

where

We assume that $K\geq X^{\frac{1}{2}}$ first. It follows from (5.17), (5.18) and Cauchy's inequality that

Suppose that $Q$ is an integer which satisfies

For the inner sum over $\ell$ in (5.19), by applying Lemma 2.4 we can derive that

where

and

Since $\lambda(d) = 0$ for $2|d$, we can assume that $(d_{1}d_{2}, 2) = 1.$ Then it follows from $d_1|k\ell+2$ and $d_2|k(\ell+q)+2$ that $(d_{1}, \ell) = (d_{2}, \ell+q) = 1.$ Hence there exists an integer $k_{0}$ that is dependent on $\ell, h, d_{1}, d_{2}$ such that the pair of conditions $k\ell+2\equiv0\; (\text{mod}\; d_{1})$ and $k(\ell+q)+2\equiv0\; (\text{mod}\; d_{2})$ is equivalent to the congruence $k\equiv k_{0}\; (\text{mod}\; [d_{1}, d_{2}])$. Thus, we have

where

For $q = 0,$ by the trivial estimate we have

For the case $q\neq 0,$ by (5.23) we get

We apply Lemma 2.3 with the following exponential pair

to derive that

Note that

and

Then we use the above two estimates, (5.21), (5.24) and (5.25) to get

for some $\gamma_{0}\in[\tau, H+1]$. We choose

Then, it is easy to check that

Substituting (5.27) into (5.26), we obtain

which implies that

When $K < X^{\frac{1}{2}}$, we can represent $W(K)$ as follows:

Now we have that $\frac{X}{K}\gg X^{\frac{1}{2}},$ then, we may proceed as in (5.19)–(5.28) but with roles of $k$ and $\ell$ reversed. Thus we can again derive the estimate (5.28). Consequently, we obtain

To bound $S_{22}$, we use the same methodology as for $S_{3}$ to derive that

Now combining (5.12), (5.15), (5.16), (5.29) and (5.30) and from the fact that $\gamma_{0}\in[\tau, H+1]$, we find that

Therefore, from condition (2.1) we conclude that if $2 < c < \frac{990}{479}$ then

With the help of (5.3)–(5.5) and (5.31), we finally obtain that

holds for $2 < c < \frac{990}{479}$, and the proof of Lemma 5.3 is completed. □

Lemma 5.4. Suppose that $2 < c < \frac{990}{479}$. Then we have

Proof. Let $G(\alpha) = \overline{L(\alpha)}|L(\alpha)|^3$. We have

From (5.32) and Cauchy's inequality, we get

Now

By the trivial bound $\mathcal{T}(\alpha, X)\ll X\log X$ and Lemma 5.3, we have

From Lemmas 5.1 and 5.3, we obtain

where (ⅲ) of Lemma 4.2 is used. It follows from (5.34)–(5.36) that

Combining (5.33), (5.37) and (ⅲ) of Lemma 4.3, we get

Now Lemma 5.4 follows from (5.38). □

We are now in a position to estimate $\Gamma_{12}$ and $\Gamma_{62}$. By Hölder's inequality and Lemma 5.4 we find that

Similarly, for $\Gamma_{62}$ we have

6.   Proof of Theorem 1.1

Proposition 6.1. We have

Proof. It follows from (4.11), (4.12) and Lemma 2.1(3) that

where the following trivial estimate is used:

□

Now according to (3.9), (5.39), (5.40) and Proposition 6.1, we obtain

which implies that $\Gamma > 0$ for a sufficiently large natural number $N.$ Then, (1.3) would have a solution in primes $p_1, p_2, p_3, p_4, p_5$ satisfying

Suppose that $p_i+2$ has $l$ prime factors, counted with multiplicity. From (6.1) and the condition $\frac{X}{2} < p_i\leq X$ we see that

Then, $l\leq \eta^{-1}.$ This means that $p_{j}+2$ has at most $\left[\frac{6227}{3960-1916c}\right]$ prime factors counted with multiplicity. Now Theorem 1.1 is proved.

Use of AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the anonymous referees for many useful comments on the manuscript.

Conflict of interest

The author declares no conflict of interest.