Abstract
For any real number y, let [y] be the largest integer not exceeding y. Petrov and Tolev conjectured that there exists a constant \(c_{0}>1\) such that if \(1<c<c_{0}\), then every sufficiently large natural number N can be represented as
where p is a prime and m is a natural number having at most 2 prime factors. And, they proved that when c is close to 1, specifically when \(1<c\le 1485/1484=1.00067\dots ,\) every sufficiently large natural number N can be represented as \(N=[p^{c}]+[m^{c}]\) with m having at most 53 prime factors. In this paper, we show that if \(1<c\le 1.0198,\) then every sufficiently large natural number N can be written as \(N=[p^{c}]+[m^{c}],\) where p is a prime and m is a natural number having at most 10 prime factors. This improves the result of Petrov and Tolev.
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1 Introduction
The famous Goldbach conjecture states that any even number greater than 2 can be written as the sum of two prime numbers. Let \(P_{r}\) denote an almost-prime with at most r prime factors counted with multiplicity. In 1966, Chen [2] announced his remarkable theorem–Chen’s theorem: every sufficiently large even integer N can written as
where and in what follows p, with or without subscript, is a prime. And the detail was published in [3].
The ancient Waring problem says that for every natural number \(k\ge 2\) there exists a positive integer \(s=s(k)\) such that every natural number is a sum of at most s kth powers of natural numbers. In 1934, Segal [12] generalized the Waring problem to fractional exponents. And, he showed that for any fixed real number \(c>1,\) there exists a positive integer \(s=s(c)\) such that every sufficiently large natural number N can be written as
where \(x_{1},x_{2},\dots ,x_{s}\) are non-negative integers. On the other hand, some mathematicians consider that how large c can be for the fixed \(s\ge 2\). In 1973, Deshouillers [4] proved that if \(1<c<4/3,\) then every sufficiently large natural number N can be represented as
where \(x_{1}\) and \(x_{2}\) are non-negative integers. Later, the domain of c was improved to \(1<c<55/41\) and \(1<c<3/2,\) respectively by Gritsenko [7] and Konyagin [9]. In addition, in 2009, Kumchev [10] proved that if \(1<c<16/15,\) then every sufficiently large natural number N can be represented as
where m is a positive integer. Recently, the range of c obtained by Kumchev was improved by Yu [14] to \(1<c<11/10.\) Furthermore, Petrov and Tolev [11] proved that if \(1<c<29/28,\) then every sufficiently large natural number N can be represented as (1.1) with m is an almost prime with at most \([52/(29-28c)]+1\) prime factors. Inspired by Chen’s theorem, Petrov and Tolev [11] proposed the following interesting conjecture:
Conjecture 1.1
There exists a constant \(c_{0}>1\) such that if \(1<c<c_{0}\), then every sufficiently large natural number N can be represented as
In the present paper, we improve the results of Petrov and Tolev when c close to 1. And, we state our theorem in the following.
Theorem 1.1
Suppose that \(1<c\le 1.0198.\) Then every sufficiently large natural number N can be represented as
where p is a prime and m is an almost prime with at most 10 prime factors.
Remark 1.1
While our method does not yield a general formula for the number of prime factors of m in terms of c, for every specific \(c\in (1, 29/28),\) one can apply our method to get an improvement to Petrov and Tolev’s result.
2 Notation and Preliminaries
From now on, let N be a sufficiently large natural number and
be the positive real numbers. Put
Put \(e(y)= e^{2\pi iy}.\) As usual, \(\mu (n)\) denotes the Möbius function. Let \(\rho (t)=\frac{1}{2}-\{t\}\), where \(\{t\}\) is the fractional part of t. Define \((\xi _{d}^{+})\) and \((\xi _{d}^{-})\) the upper bound and lower bound beta-sieves of level D respectively (see Chapter 11 of [5]), for which we have
For \(z\ge 2,\) define
Then, by Theorem 11.12 of Friedlander and Iwaniec [5], we have
and
where F(s) and f(s) are the standard upper and lower bound functions of the linear sieve, and
Lemma 2.1
For F(s) and f(s), We have
Proof
See [8, (3.11) and (3.12)]. \(\square \)
We denote
where \(\eta =\eta (c)>0\) is a constant. Let
Lemma 2.2
Let
Then we have
Proof
See (23), (24), (71), and (73) of [11]. \(\square \)
Lemma 2.3
(Vaaler’s theorem) For each \(H\ge 2\) there are numbers \(c_{h}\), \(1\le h\le H,\) and \(d_{h},\) \(0\le h\le H,\) such that
where
and
Proof
See [13].
Lemma 2.4
(Van der Corput’s Theorem) Suppose that \(\vartheta \) is a real valued function with two continuous derivatives on interval I. Suppose also that there is some \(\lambda >0\) such that
on I. Then
Proof
See [6, Theorem 2.2]. \(\square \)
3 A Key Mean Estimation
In this section, we prove a mean estimation similar to Lemma 2.2, which plays a crucial role in the proof of Theorem 1.1.
From now on, we take \(z=N^{\frac{1}{200}}.\) Let
where
By the prime number theorem, we have
Taking \(x=(1-\delta ^{c})^{\theta }N^{\theta }+1\) in [1, (4.29)] and by some routine arguments we get that
where
To compute the bound \(c_{k}\) we used the Mathematica technical computing software. For example, we use the following code to calculate \(c_{11}.\)
NIntegrate [(Log[t9 − 1]/t9)*(1/t8)*(1/t7)*(1/t6)*(1/t5)*(1/t4)*(1/t3)*(1/t2)*(1/t1), {t1, 10, 199}, {t2, 9, t1 − 1}, {t3, 8, t2 − 1}, {t4, 7, t3 − 1}, {t5, 6, t4 − 1}, {t6, 5, t5 − 1}, {t7, 4, t6 − 1}, {t8, 3, t7 − 1}, {t9, 2, t8 − 1}]
In fact, the estimate of \(c_{k}\) for \(k\ge 15\) has already been given in [1, (4.30)]. Whatever, we have
Hence, we have
Proposition 3.1
Let
If \( k\ge 2,\) then we have
Proof
Let \(Z\ge 2\) be any fixed integer. From page S43 and S44 of [11], we know that there exists a series of periodic functions \(g_{s}(t), s=0,1,\dots ,2Z-1\) with a period of 1 and has the following properties:
and
Furthermore, we have
where
By Lemma 2.3, we can write
where
and
Let
Changing the order of summation together with Lemma 2.3, we obtain
and
Hence, by (3.3), we obtain
We choose
Combining (3.10), (3.11) and (3.12), we obtain
Now, we consider the sum \(W_{j}(v)\). From (3.7) it follows that
where
By (3.2), (3.8) and (3.9). we have
where \(A=(1-(2\delta )^{c})^{\theta },\) \(B=(1-\delta ^{c})^{\theta }\) and
Let \(\vartheta _{n}(x)=nx^{c}.\) It is easy to verify that \(\vartheta _{n}^{\prime \prime }(x)\asymp nN^{1-2\theta }\) uniformly for \(AN^{\theta }-1\le x<BN^{\theta }+1.\) Hence, by Lemma 2.4, we get
We assume that
Then by (3.14), (3.15) and (3.16), we obtain
Now, we consider the sums \(W_{j}^{(s)}(v)\) for \(1\le s\le 2Z-1.\) From (3.5) we know that \(g_{s}(m^{c})\) vanishes unless \(\{m^{c}\}\in [(s-1)/(2Z),(s+1)/(2Z)].\) Hence, the only summands in the sums \(W_{s}(v)\) are those for which
And in this case we have
and so
Hence, we have
where
Thus, by (3.17), we get
where
By (3.3), (3.5) and (3.6), we have
Now, we have
Take
Obviously, we have \(vN^{2\theta -1}\ll N^{\theta }(\log N)^{3}.\) So we can rewrite (3.19) as
Combining (3.13) and (3.20), we obtain
We choose Z such that
Hence, we have
Now, we consider the sums \(V_{j}^{(s)}(h/d).\) By (3.8) and (3.18), we have
where
Inserting (3.22) into (3.21), we get
Recall the definition of \(\mathcal {M}_{k}\) in (3.2). Let \(k\ge 2\) and \(n=p_{1}\cdots p_{k}\in \mathcal {M}_{k}.\) By some routine arguments, we can rewrite n as \(n=rs\) with
In fact, it is easy to see that \(U\Big (T,n,\frac{h}{d}\Big )\) is a summation similar to (121) in [11]. Through the same argument as Sect. 3.6 of [11], there is almost no need for adjustment, and we can get that if
then
So, we have
This completes the proof. \(\square \)
4 The Proof of Theorem 1.1
To prove the theorem, we consider the lower bound of the sum
By the trivial inequality
From (6), (17), (18) and (19) in [11], we know that
where \(\Sigma _{j}, j=0,1\) are defined by (2.4) and
with
and
By Lemma 2.2 and Proposition 3.1, we can take
So we have
From the definition of P and the prime number theorem, we obtain
Hence, by Lemma 2.2 and the fact
we get
Obviously, we have
where
From (2.1), we find
By exchanging the order of summation, we obtain
By the identity
we have
Combining (4.6), (4.7) and the identity
we obtain
where \(R^{(k)}_{j}, j=0,1\) are defined by (3.1) and
where
By (3.4), (4.3), (4.5), (4.8), (4.9) and Proposition 3.1, we have
From (4.1), (4.4) and (4.10), as long as
we can deduce that
which leads to the theorem. Recall that \(\delta =10^{-9}.\) By (4.2) and Lemma 2.1, one can use the software Mathematica to run the following code, which shows that \(L[1.0198]>0.0017884.\)
F[x_] := Piecewise [{{(2E\(^\wedge \)EulerGamma)/x, 0 < x \(<=\) 3}, {((2E\(^\wedge \)EulerGamma)/x) * (1 + NIntegrate [Log [t \(-1\)]/t, {t, 2, x \(-1\)}]), 3 \(<=\) x < 5}, {((2E\(^\wedge \)EulerGamma)/x) * (1 + NIntegrate [ Log[t \(-1\)]/t, {t, 2, x \(-1\)}] + NIntegrate [(Log [t \(-1\)]/(t * u)) * Log [(u \(-1\))/(t \(+1\))], {t, 2, x \(-3\)}, {u, t+2, x \(-1\)}]), 5 \(<=\) x<7}}]
f[x_] := Piecewise [{{((2E\(^\wedge \)EulerGamma)/x) * Log [x \(-1\)], 2 \(<=\) x \(<=\) 4}, {((2E\(^\wedge \)EulerGamma) /x) * (Log [x \(-1\)] + NIntegrate [Log [u \(-1\)]/(t * u), {t, 3, x \(-1\)}, {u, 2, t \(-1\)}]), 4 \(<=\) x < 6}, {((2E\(^\wedge \)Eu lerGamma)/x) * (Log [x \(-1\)] + NIntegrate [Log [u \(-1\)]/(t * u), {t, 3, x \(-1\)}, {u, 2, t \(-1\)}] + NIntegrate [ (Log [t \(-1\)]/(t * u)) * Log [(u \(-1\))/(t + 1)] * Log[x/(u + 2)], {t, 2, x \(-4\)}, {u, t + 2, x \(-2\)}]), 6 \(<=\) x \(<=\) 8}} ]
L[x_] := ((1 − (2 * 10\(^\wedge \)(− 9))\(^\wedge \)x)\(^\wedge \) (1/x \(-1\))) * f [(200 * (29–28x))/(26x)] − (10\(^\wedge \)9) * (2 * 10\(^\wedge \)(− 9))\(^\wedge \) (1 −x) ((1–10\(^\wedge \)(− 9x))\(^\wedge \) (1/x) − (1 − (2 * 10\(^\wedge \)(− 9))\(^\wedge \)x)\(^\wedge \) (1/x)) F [(200 * (29–28x))/(26x)] * 0.840654
From [5, Chapter 11], we know that f(s) is an increasing function and F(s) is a decreasing function. By a trivial argument, we can conclude that L is a decreasing function about c, which deduces that if \(1<c<1.0198,\) then \(L>0.0017884\). This completes the proof of the theorem.
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Acknowledgements
The authors would like to thank the editor and referee for their time extended on the manuscript. The first author is supported by Project funded by China Postdoctoral Science Foundation No. 2023M732666. The second author is supported by The National Natural Science Foundation of China No. 11771333.
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Zhou, GL., Cai, Y. On a Conjecture of Petrov and Tolev Related to Chen’s Theorem. Bull. Malays. Math. Sci. Soc. 47, 124 (2024). https://doi.org/10.1007/s40840-024-01708-1
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DOI: https://doi.org/10.1007/s40840-024-01708-1