1. Introduction
The Bertrand-Chebyshev’s theorem states that for any positive integer n, there always exists a prime number p such that
. Pafnuty Chebyshev proved this in 1850 [1] . In 2006, M. El Bachraoui [2] extended the theorem and proved that for any positive integer n, there exists a prime number p such that
. In 2011, Andy Loo [3] proved that when n ≥ 2, there are prime numbers in the interval (3n, 4n). In 2013, Vladimir Shevelev et al. [4] proved that when the integer k ≤ 100,000,000, only k = 1, 2, 3, 5, 9, 14, for all n ≥ 1, the interval (kn, (k + 1)n) contains prime numbers. This raises the question: for all k ≥ 1, under what conditions does the interval (kn, (k + 1)n) contain prime numbers? Previously, the author partially answered this question in the paper [5] by
analyzing the binomial coefficients
where λ is a positive integer. In that
paper, the author proved that when
, i.e., when
and
, there exists at least a prime number p such that
. In this article, we will use the same method to complete the entire work on this problem. We will prove that when
and
, there exists at least a prime number p such that
. Then, by converting λ to k, we prove that for two positive integers n and k, when
, there is always at least a prime number p such that
. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1.
We will use the same definition and concepts from [5] in this section and in section 2. In section 3, we will prove that for λ from 3 to 26, when
, there exists at least a prime number p such that
. In section 4, we will convert λ to k, and complete this article.
Definition:
denotes the prime number factorization operator of the integer expression
. It is the product of the prime numbers in the decomposition of
in the range of
. In this operator, p is a prime number,
and
are real numbers, and
.
It has some properties:
It is always true that
(1.1)
If there is no prime number in
within the range of
, then
, or vice versa, if
, then there is no prime number in
within the range of
. (1.2)
For example, when
and
,
. No prime number 13 or 11 is in
within the range of
.
If there is at least one prime number in
in the range of
, then
, or vice versa, if
, then there is at least one prime number in
within the range of
. (1.3)
For example, when
and
,
. A prime number 17 is in
within the range of
.
Let
be the p-adic valuation of n, the exponent of the highest power of p that divides n.
We define
by the inequalities
, and determine the p-adic valuation of
.
because for any real numbers
and
, the expression of
is 0 or 1.
Thus, if p divides
, then
, or
(1.4)
If
, then
. (1.5)
Let
be the number of distinct prime numbers less than or equal to 𝑛. Among the first six consecutive natural numbers are three prime numbers 2, 3, and 5. Then, for each additional six consecutive natural numbers, at most one
can add two prime numbers,
and
. Thus,
. (1.6)
From the prime number decomposition, when
,
When
,
Thus,
since all prime numbers in
do not appear in the range of
.
If
, and there is a prime number p in
, then
. From (1.5),
. Thus, if
, every prime number in
has a power of 0 or 1. (1.7)
Referring to (1.5),
. It has been proven [6] that for
,
.
When
,
, then for
,
.
Referring to (1.4) and (1.6),
.
Thus, for
and
,
(1.8)
2. Lemmas
Lemma 1: For
and
,
(2.1)
Proof:
Let a real number
, and
; then,
.
Thus,
is a strictly decreasing function for
.
Since
, and
, for
, we have
.
Let
, then
(2.1.1)
In (2.1.1), for
,
Using the formula:
, we have
Thus, for
,
.
Since
is a positive number for
,
.
Thus
is a strictly deceasing function for
. Since
and
, for
,
(2.1.2)
Since for
,
has a lower bound of 4 and
has an upper bound of 3.375,
When
, we have
(2.1.3)
When
and
,
(2.1.4)
(2.1.5)
Since
is a positive number for
, referring to (2.1.4) and (2.1.5), when
multiplies both sides of (2.1.3), we have
Thus,
when
and
. (2.1.6)
By induction on n, when
, if
is true for n, then for
,
Notice
, and
because
. Thus,
(2.1.7)
From (2.1.6) and (2.1.7), we have for
and
,
Referring to (1.8), for
and
,
.
Then,
.
Since when
and
, then
and
.
Thus, Lemma 1 is proven.
Lemma 2: When
,
is an increasing function with respect to the product of λn and with respect to n. (2.2)
Proof:
Referring to (2.1.2), when
,
. From (2.1), when
and
,
, and
.
Thus,
. (2.2.1)
Let
and
both be real numbers. When
,
(2.2.2)
where
when
.
Thus,
is a strictly increasing function for
.
When
,
. Thus, for
,
. Then,
.
Thus,
is a strictly increasing function for
.
From (2.2.2), when
,
. Thus, when
and
, then
is an increasing function respect to the product of xy. (2.2.3)
(2.2.4)
where
When
, then
When
,
(2.2.5)
When
, then
Thus,
then
When
,
, and from (2.2.5),
.
Then
.
Thus, when
,
is a strictly increasing function.
When
, since
,
is an increasing function respect to xy.
When
,
.
When
,
.
Thus, when
,
, and it is an increasing function with respect to x and to the product of xy, then, from (2.2.4),
.
Thus, when
,
is an increasing function with respect to x. (2.2.6)
Referring to (2.2.3) and (2.2.6), when
, then
,
is an increasing function with respect to the product of xy and with respect to x.
Let
and
. Then when
,
is an increasing function with respect to the product of λn and with respect to n.
Thus, Lemma 2 is proven.
Lemma 3: When
and
,
is a strictly increasing function respect to n. (2.3)
Proof:
Referring to (2.1), when
and
,
Let a real number
. When
, then
(2.3.1)
When λ is an integer constant in the range of
,
where
for
and
; then,
is a strictly increasing function respect to x.
When
and
,
. The calculations are below.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
From (2.3.1), when
and
,
.
Since
, when
and
,
.
Thus, when
and
,
is a strictly increasing function respect to x.
Let
and
,
is a strictly increasing function respect to n. (2.3.2)
From (2.2.1), when
and
,
. Referring to (2.2), when
and
,
is an increasing function with respect to the product of λn and with respect to n.
When
and
, since
and
,
is an increasing function with respect to the product of λn and with respect to n. (2.3.3)
From (2.3.2) and (2.3.3), when
and
,
is an increasing function with respect to n.
Thus, Lemma 3 is proven.
Lemma 4: When
and
, if
, then there exists at least a prime number p such that
. (2.4)
Proof:
Let integers
.
From (2.3), when
and
,
is an increasing function respect to n.
When
, if
, then
, and thus,
; then from (2.1),
, and
. (2.4.1)
Note that
since
.
From (1.7), when
, every prime number in
has a power of 0 or 1; when
, every prime number in
has a power of 0 or 1. (2.4.2)
In
, for every distinct prime number p in these ranges, the numerator
has the product of
. The denominator
also has the same product of
. Thus, they cancel each other in
.
Referring to (1.2),
.
Thus,
. (2.4.3)
is the product of (λ − 1) sectors from
to
.
Each of these sectors is the prime number factorization of the product of the consecutive integers between
and
.
Referring to (2.4.3), when
, then
.
Referring to (1.1),
. Thus, when
, at least one of the sectors is greater than one in
.
Let
be such a sector and let
where
from (2.4.3). Thus, when
,
. (2.4.4)
Thus,
contains all the factors of
in
.
These factors make up all the consecutive integers in the range of
in
. Thus,
contains
.
Referring to the definition, all prime numbers in
in the ranges of
and
do not contribute to
, nor does i for
. Only the prime numbers in the prime factorization of
in the range of
present in
. Since
is the product of all the consecutive integers in this range,
.
Referring to (2.4.4),
. (2.4.5)
Referring to (2.4.1), (2.4.3), (2.4.4), and (2.4.5), when
and
,
if
, then
and
;
when
, then
.
Thus, when
and
, if
, then
, referring to (1.3), there exists at least a prime number p such that
.
Thus, Lemma 4 is proven.
Lemma 5: When
, there exists at least a prime number p such
that
. (2.5)
Proof:
Referring to (2.2), when
,
is an increasing function with respect to the product of λn and with respect to n.
Let integers
, and
. Then
.
When
,
.
From (2.2), when
,
and
since
is an increasing function with respect to the product of λn. When
, then
and
since
is also an increasing function with respect to n.
Referring to (2.2.1), when
,
; and
. (2.5.1)
Referring to (2.4.2), when
, every prime number in
and in
has a power of 0 or 1. (2.5.2)
Referring to (2.5.1), (2.4.3), (2.4.4), and (2.4.5), when
,
, and
, then
; referring to (1.3), there exists at least a prime number p such that
.
Thus, Lemma 5 is proven. It was proven in [ [5] , pp 1324-1329] with more details.
3. A Prime Number between (λ − 1)n and λn When 26 ≥ λ ≥ 3 and n ≥ λ − 2
Proposition 1: For
, when
, there exists at least a prime number p such
that
. (3.1)
Proof:
Referring to (2.3) for
, when
,
is a strictly increasing function on
.
Let
. When
and
,
.
Thus, for
, when
,
and
; then, referring to (2.4), there exists at least a prime number p such that
.
For
, when
, Table 1 shows that there exists a prime number p such that
.
Thus, (3.1) is proven.
Proposition 2: For
, when
, there exists at least a prime number p such that
. (3.2)
Table 1. For
and
, there is a prime number p such that
.
Proof:
From (2.3), for
, when
,
is a strictly increasing function on
.
Let
. When
and
,
. The calculations are below.
When
,
.
When
,
.
Thus, for
, when
,
and
; then, referring to (2.4), there exists at least a prime number p such that
.
For
, when
, Table 2 shows that there exists a prime number p such that
.
Thus, (3.2) is proven.
Proposition 3: For
, when
, there exists at least a prime number p such that
. (3.3)
Proof:
Referring to (2.3), for
, when
,
is a strictly increasing function with respect to
.
Let
. When
and
,
.
Table 2. For
and
, there is a prime number p such that
The calculations are below.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
Thus, for
, when
,
and
; then, referring to (2.4), there exists at least a prime number p such that
.
For
, when
, Table 3 shows that there exists a prime number p such that
.
Thus, (3.3) is proven.
Proposition 4: For
, when
, there exists at least a prime number p such that
. (3.4)
Proof:
Referring to (2.3) for
, when
,
is a strictly increasing function with respect to
. Let
. When
and
, then
and
.
For
and
, then
. It can be seen from Table 4 that the values of
as well as
are all greater than 1.
(Detailed calculations are in Appendix.)
Thus, for
, when
,
; then, referring to (2.4), there exists at least a prime number p such that
.
For
and
, Table 5 shows that there exists a prime number p such that
.
For
and
, Table 6 shows that there exists a prime number p such that
.
Thus, (3.4) is proven.
Combining (3.1), (3.2), (3.3), and (3.4), when
and
, there exists at least a prime number p such that
. (3.5)
4. A Prime Number between kn and (k + 1)n When n ≥ k − 1
Proposition 5: For two positive integers n and k, when
, there exists at least a prime number p such that
. (4.1)
Table 3. For
and
, there is a prime number p such that
.
Table 4. When
and
, then
.
Table 5. For
and
, there is a prime number p such that
.
Table 6. For
and
, there is a prime number p such that
.
Proof:
Referring to (2.5), when
, there exists at least a prime number p such that
. This statement is the same as that when
and
, there exists at least a prime number p such that
.
Referring to (3.5), when
and
, there exists at least a prime number p such that
.
Thus, when
and
, there is at least a prime number p such that
.
Let integer
, then for
and
, when
, there exists at least a prime number p such that
. (4.2)
The Bertrand-Chebyshev’s theorem points out that for
and
, there exists at least a prime number p such that
. (4.3)
From (4.2) and (4.3), we can conclude that for
and
, when
, there exists at least a prime number p such that
. Thus, Proposition 5 becomes a theorem, Theorem 4.1, and Bertrand-Chebyshev’s theorem is a special case of this theorem.
In the field of prime number distribution, an important theorem is the prime number theorem,
, where
is the number of distinct
prime numbers less than or equal to a natural number N. The prime number theorem provides the approximate number of prime numbers relative to the natural numbers, while Theorem 4.1 shows that when
, the prime number exists in the interval between kn and (k + 1)n, that is, Theorem 4.1 provides the approximate locations of prime numbers among natural numbers. Using this theorem, Legendre’s conjecture [7] and several other conjectures can be easily proven. The method of proving Theorem 4.1 can also help to study and solve some difficult problems in number theory such as the other three Landau problems [8] .
Appendix
Calculation of
when
and
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.
When
,
.