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arXiv:1110.3465v19 [math.GM] 20 May 2016
Acta Mathematica Sinica, English Series
Sep., 201x, Vol. x, No. x, pp. 1–
Published online: August 15, 201x
DOI: 0000000000000000
Http://www.ActaMath.com
On the representation of even numbers as the sum and difference of
two primes and the representation of odd numbers as the sum of an
odd prime and an even semiprime and the distribution of primes in
short intervals
Shan-Guang Tan
Zhejiang University, Hangzhou, 310027, China
tanshanguang@163.com
Abstract Part 1. The representation of even numbers as the sum of two odd primes and the
distribution of primes in short intervals were investigated in this paper. A main theorem was proved.
It states: There exists a finite positive number n 0 such that for every number n greater than n 0 , the
even number 2n can be represented as the sum of two odd primes where one is smaller than
√ 2 n and
another is greater than 2n −
√ 2 n.
The proof of the main theorem is based on proving the main assumption ”at least one even number
greater than 2n 0 can not be expressed as the sum of two odd primes” false with the theory of linear
algebra. Its key ideas are as follows (1) For every number n greater than a positive number n 0 , let
Qr = {q 1 , q 2 , · · · , qr } be the group of all odd primes smaller than
√ 2 n and gcd(qi, n) = 1 for each
qi ∈ Qr where qr <
√ 2 n < qr+1. Then the even number 2n can be represented as the sum of an
odd prime qi ∈ Qr and an odd number di = 2n − qi. (2) Based on the main assumption, all the odd
numbers di should be composite, then a group of linear algebraic equations can be formed and with
the solutions qi ∈ Qr. (3) When a contradiction between the expectation and the actual results of the
solutions is obtained, the main assumption is proved false so that the main theorem is proved.
It has been found that n 0 = 31, 637. Based on the main theorem and by verifying for n ≤ n 0 , it was
proved that for every number n greater than 1, there are always at least one pair of primes p and q
which are symmetrical about the number n so that even numbers greater than 2 can be expressed as
the sum of two primes. Hence, Goldbach’s conjecture was proved.
Also based on the main theorem, some theorems of the distribution of primes in short intervals were
proved. By these theorems, the Legendre’s conjecture, the Oppermann’s conjecture, the Hanssner’s
conjecture, the Brocard’s conjecture, the Andrica’s conjecture, the Sierpinski’s conjecture and the
Sierpinski’s conjecture of triangular numbers were proved and the Mills’ constant can be determined.
Part 2. The representation of odd numbers as the sum of an odd prime and an even semiprime, and the
distribution of primes in short intervals were investigated in this paper. A main theorem was proved.
It states: There exists a finite positive number n 0 such that for every number n greater than n 0 , the
odd number 2n + 1 can be represented as the sum of an odd prime p and an even semiprime 2q where
q is an odd prime smaller than
√ 2 n and p is greater than 2n + 1 − 2
√ 2 n.
The proof of the main theorem is based on proving the main assumption ”at least one odd number
greater than 2n 0 + 1 can not be expressed as the sum of an odd prime and an even semiprime where
Received x x, 201x, accepted x x, 201x
2 Shan-Guang Tan
n 0 ≥ 2” false with the theory of linear algebra. Its key ideas are as follows (1) For every number n
greater than a positive number n 0 , let Qr = {q 1 , q 2 , · · · , qr } be the group of all odd primes smaller
than
√ 2 n and gcd(qi, 2 n + 1) = 1 for each qi ∈ Qr where qr <
√ 2 n < qr+1. Then the odd number
2 n + 1 can be represented as the sum of an even semiprime 2qi and an odd number di = 2n + 1 − 2 qi
where qi ∈ Qr. (2) If all the odd numbers di are composite, then a group of linear algebraic equations
can be formed and with the solutions qi ∈ Qr. (3) When a contradiction between the expectation and
the actual results of the solutions is obtained, the main assumption is proved false so that the main
theorem will be proved.
It has been found that n 0 = 19, 875. Based on the main theorem and by verifying for n ≤ n 0 , it was
proved that for every number n greater than 2, there are always at least one pair of primes p and q
so that all odd integers greater than 5 can be represented as the sum of an odd prime and an even
semiprime. Therefore, Lemoine’s conjecture was proved.
Also based on the main theorem, some theorems of the distribution of primes in short intervals were
given out and proved. By these theorems, the Legendre’s conjecture and the Andrica’s conjecture were
proved and the Mills’ constant can be determined.
Part 3. It was proved in this paper that:
(1) Let R 2 (x) denote the number of odd prime pairs p and x − p for 3 ≤ p ≤ x/2. Then for large even
numbers x, a positive number N always exists such that the inequalities
R
∗ 2 (x, N^ )^ ≤^ R^2 (x)^ < R
∗ 2 (x, N^ + 1)
hold where
R
∗ 2 (x, N^ ) =^ αx
x
log
2 x
∑^ N
n=
n!
log
n− 1 x
,
and R 2 (x) is asymptotically equal to
R 2 (x) = αxLi 2 (x) + O(
√ x log x)
where αx is a positive constant dependent of x(mod 6).
(2) Let R 3 (x) denote the number of odd prime pairs p and x − 2 p for 3 ≤ p ≤ (x − 3)/2. Then for large
odd numbers x, a positive number N always exists such that the inequalities
R
∗ 3 (x, N^ )^ ≤^ R^3 (x)^ < R
∗ 3 (x, N^ + 1)
hold where
R
∗ 3 (x, N^ ) =^ αx
x
log 2 x
∑^ N
n=
n!
log n− 1 x
,
and R 3 (x) is asymptotically equal to
R 3 (x) = αxLi 2 (x) + O(
√ x log x)
where αx is a positive constant dependent of x(mod 6).
Part 4. It is proved in this paper that there exists a finite positive number x 0 such that for every
positive number x = pn+1 + 1 greater than x 0 , there is asymptotically
pi+1 − pi = O(log
2 pi)
and
lim sup i→∞
pi+1 − pi
log 2 pi
= c
4 Shan-Guang Tan
over the integers, and (3) as m runs over the positive integers the numbers f (m) should not share a
common factor greater than 1, and f (x) can be converted to a new polynomial f (x) such that the
integer coefficients of the new polynomial f (x) satisfy the above three necessary conditions and the two
conditions: (4) the inequality (n + 1)
4 ≤ ¯ f (1)/ log
(^2) ¯ f (1) holds, and (5) gcd( ¯ f (1), a 0 ) = 1, then f (m) is
prime for infinitely many positive integers m.
Part 9. The number of Bunyakovsky primes not greater than f (x) was investigated and a main theorem
was given out and proved in this paper, which states: For every number x greater than x 0 , there are
always Bunyakovsky primes in the region [x 1 , x] where f (x 1 − 1) ≤
√ f (x) < f (x 1 ). Let denote the
number of Bunyakovsky primes in the region [x 1 , x] by τ (x) and the number of Bunyakovsky primes
not greater than f (x) by πf (x), respectively. Then
τ (x) ≥ αf
x
log x
√ x)
√ x
2 log x
≥ 1
where αf and βf are positive constants dependent of the polynomial f (x).
Based upon the main theorem, it was proved that:
(1) The number of Bunyakovsky primes not greater than f (x) satisfies
π
∗ f (x, N^ )^ ≤^ πf (x)^ < π
∗ f (x, N^ + 1)
where N is a positive number and
π
∗ f (x, N^ ) =^ αf
x
log x
∑^ N
n=
n!
log
n x
.
(2) For any positive number x ≥ 10
5 a positive number N always exists such that the number of
Bunyakovsky primes not greater than f (x) satisfies
πf (x) − π
∗ f (x, N^ )^ < αf (x^ log^ x)
1 / 2 .
(3) For large numbers x, πf (x) is asymptotically equal to
πf (x) = αf Li(x) + O(
√ x log x).
Part 10. The representation of even numbers as the sum of one odd prime and the product of a group
of distinct odd primes was investigated in this paper. A basic theorem was proved. It states: For any
finite positive number s > 1, there exists a finite positive number ns such that for every number n
greater than ns, the even number 2n can be represented as the sum of one odd prime and the product
of a group of s distinct odd primes where the prime is smaller than log
2 θ (2n) and the product is greater
than 2n − log
2 θ (2n) where 1 ≤ θ and log
2 θ (2n) <
√ 2 n.
The proof of the basic theorem is based on proving the main assumption ”at least one even number
greater than 2ns can not be expressed as the sum of one odd prime and the product of a group of s
distinct odd primes” false with the theory of linear algebra. Its key ideas are as follows (1) For every
number n greater than a positive number ns, let Qr = {q 1 , q 2 , · · · , qr } be the group of all odd primes
smaller than log
θ (2n) and gcd(qi, n) = 1 for each qi ∈ Qr where qr < log
θ (2n) < qr+1. Then the even
number 2n can be represented as the sum of the product of a group of s distinct odd primes λiqi and
an odd number di = 2n − λiqi. (2) If all the odd numbers di are composite, then a group of linear
algebraic equations can be formed and with the solutions qi ∈ Qr. (3) When a contradiction between
Representation of numbers and distribution of primes 5
the expectation and the actual results of the solutions is obtained, the main assumption is proved false
so that the basic theorem is proved.
Part 11. The representation of odd numbers as the sum of one even semiprime and the product of a
group of distinct odd primes was investigated in this paper. A basic theorem was proved. It states: For
any finite positive number s > 1, there exists a finite positive number ns such that for every number n
greater than ns, the odd number 2n + 1 can be represented as the sum of one even semiprime and the
product of a group of s distinct odd primes where the even semiprime is smaller than 2 log 2 θ (2n) and
the product is greater than 2n + 1 − 2 log 2 θ (2n) where 1 ≤ θ and log 2 θ (2n) <
√ 2 n.
The proof of the basic theorem is based on proving the main assumption ”at least one odd number
greater than 2ns + 1 can not be expressed as the sum of one even semiprime and the product of a
group of s distinct odd primes” false with the theory of linear algebra. Its key ideas are as follows (1)
For every number n greater than a positive number ns, let Qr = {q 1 , q 2 , · · · , qr } be the group of all
odd primes smaller than log
θ (2n) and gcd(qi, 2 n + 1) = 1 for each qi ∈ Qr where qr < log
θ (2n) < qr+1.
Then the odd number 2n + 1 can be represented as the sum of the product of a group of s distinct
odd primes λiqi and an even number 2di = 2n + 1 − λiqi where di is an odd number. (2) If all the
odd numbers di are composite, then a group of linear algebraic equations can be formed and with the
solutions qi ∈ Qr. (3) When a contradiction between the expectation and the actual results of the
solutions is obtained, the basic theorem will be proved.
Part 12. It is proved in this paper that for any finite positive number k ≥ 2, let f 1 (x), f 2 (x), · · · , fk (x)
be irreducible polynomials over the integers, with integral coefficients and positive leading coefficient.
Assume that the following condition holds: there does not exist any integer d > 1 dividing all the
products f 1 (m)f 2 (m) · · · fk (m), for every integer m, and the polynomial
f (x) =
∏^ k
i=
fi(x) = anx
n
n− 1
- · · · + a 1 x + a 0 = f¯ (x) + a 0
can be converted to a new polynomial f (x) such that the integer coefficients of the new polynomial
f (x) satisfy the above conditions and the following conditions: (1) a 0 > 0, an > 0 and ai ≥ 0 for
i = 1, 2 , · · · , n − 1, (4) the inequality (n + 1)
4 ≤ f¯ (1)/ log
(^2) ¯ f (1) holds, and (5) gcd( f¯ (1), a 0 ) = 1,
then there exist infinitely many natural numbers m such that all numbers f 1 (m), f 2 (m), · · · , fk (m) are
primes.
Part 13. Let κk (x) denote the number of prime k-tuples in every admissible pattern for a prime
constellation (p, p + 2c 1 , p + 2c 2 , · · · , p + 2ck− 1 ) for k ≥ 2 and 0 < c 1 < c 2 < · · · < ck− 1 where p ≤ x.
Then for large numbers x, the number κk (x) satisfies
κk (x) ≥ ak
x
log
k x
x
log
k+ x
where ak and bk are positive constants dependent of the number k.
Based upon the result, it was proved that:
(1) The number κk (x) satisfies
κ
∗ k (x, N^ )^ ≤^ κk^ (x)^ < κ
∗ k (x, N^ + 1)
where N is a positive number and
κ
∗ k (x, N^ ) =^
ak
(k − 1)!
x
log
k x
∑^ N
n=
(n + k − 1)!
log
n x
.
(2) For any small positive value ǫ(x), a pair of positive numbers x and N always exist such that the
Representation of numbers and distribution of primes 7
positive degree n and 0 < q ≤ x. It is proved in this paper that there are constants α > 0 and αf > 0
such that as x → ∞,
πf (x) =
αf
n
∫ x
2
du
log u
− 1 2 α
√ log x ).
Part 16. For any finite positive number k ≥ 1, let f 1 (x), f 2 (x), · · · , fk (x) be irreducible polynomials
over the integers, with positive leading coefficients. Assume that the following condition holds: there
does not exist any integer d > 1 dividing all the products f 1 (m)f 2 (m) · · · fk (m), for every integer m.
Let πk (x) denote the number of natural numbers m ≤ x such that all numbers f 1 (m), f 2 (m), · · · , fk (m)
are primes. It is proved in this paper that as x → ∞,
πk (x) =
αk
D
∫ (^) x
2
du
log
k u
− 1 2 α
√ log x )
where α > 0 and αk > 0 are constants, and D is the product of the degrees of the polynomials. Then
based on the prime number theorem for a group of polynomials of arbitrary degree, the prime number
theorems for twin primes and Sophie Germain primes can be proved.
Part 17. It is pointed out in this paper that there is a fatal fault in the proof of Maier’s theorem
proposed in H. Maier’s paper in 1985 such that the Maier’s theorem should be false.
With corrections of the proof of Maier’s theorem, it is proved in this paper that: Let Φ(x) :=
(log x)
λ , λ ≥ 2. Then as x → ∞ there is
lim x→∞
π(x + Φ(x)) − π(x)
Φ(x)/ log x
= 1.
Part 18. Based on a theorem for the numbers of primes in short intervals, some conjectures of distri-
bution of primes in short intervals are proved true.
Part 19. A theorem for the numbers of integers with k prime factors in short intervals was proved in
this paper, which states: Let g(x) = (log x)
λ , λ ≥ 2. Let πk (x) denote the number of positive integers
not exceeding x that can be written as the product of exactly k distinct primes. For any finite positive
number k ≥ 1, there exists a finite positive number xk such that for every positive number x greater
than xk , the number of integers with k prime factors in the interval (x, x + g(x)] is asymptotically equal
to
πk (x + g(x)) − πk (x) =
g(x)(log log x)
k− 1
(k − 1)! log x
g(x)(log log x)
k− 1
(k − 1)! log x
).
Part 20. Problems on irreducible polynomials were investigated in this paper. Some theorems were
proved as follows
(1) For a polynomial fn(X) ∈ Z(X) and
fn(X) = anX
n
n− 1
1
where a prime p|an, if fn− 1 (X) is irreducible over Q(X) and ¯ fn− 1 (X) = fn− 1 (X) where ¯ fn− 1 (X) ≡
fn− 1 (X)(mod p), then fn(X) is irreducible over Q(X).
(2) For any positive integer n, the polynomial
sn(x) = 1 +
n ∑
k=
pk+1x
k = 1 + p 2 x + p 3 x
2
n
is irreducible over the field Q of rational numbers.
(3) For any positive integer n, the polynomial
sn(x) =
n ∑
m=
(m + 1)x
m = 1 + 2x + 3x
2
n
8 Shan-Guang Tan
is irreducible over the field Q of rational numbers. Moreover, sn(x) is reducible modulo an odd prime
p if and only if n = kp − 1 or n = kp − 2, where k is a positive integer greater than 1.
(4) For any positive integer n, the polynomial
sn(x) =
n ∑
m=
(m + 1)x
m = 1 + 2x + 3x
2
n
is irreducible over the field Q of rational numbers. Moreover, sn(x) is reducible modulo any prime not
greater than pm if and only if n = κ(pm#) − 1 or n = κ(pm#) − 2, where κ is a positive integer, pm is
the mth prime and pm# = p 1 p 2 · · · pm. Furthermore sn(x) is reducible modulo any prime if and only
if n = κ(pm#) − 1 or n = κ(pm#) − 2 when m → ∞ such that n → ∞.
Keywords number theory, distribution of primes, Goldbach’s conjecture, Lemoine’s conjecture, Leg-
endre’s conjecture, Oppermann’s conjecture, Brocard’s conjecture, Andrica’s conjecture, Cram´er’s con-
jecture, Polignac’s conjecture, Bunyakovsky’s conjecture, Dickson’s conjecture, Schinzel’s hypothesis
H, irreducible polynomial
MR(2010) Subject Classification 11A41, 11C08, 12E
Part I
On the representation of even numbers
as the sum of two primes and the
distribution of primes in short intervals
Abstract The representation of even numbers as the sum of two odd primes and the distribution of
primes in short intervals were investigated in this paper. A main theorem was proved. It states: There
exists a finite positive number n 0 such that for every number n greater than n 0 , the even number 2n
can be represented as the sum of two odd primes where one is smaller than
√ 2 n and another is greater
than 2n −
√ 2 n.
The proof of the main theorem is based on proving the main assumption ”at least one even number
greater than 2n 0 can not be expressed as the sum of two odd primes” false with the theory of linear
algebra. Its key ideas are as follows (1) For every number n greater than a positive number n 0 , let
Qr = {q 1 , q 2 , · · · , qr } be the group of all odd primes smaller than
√ 2 n and gcd(qi, n) = 1 for each
qi ∈ Qr where qr <
√ 2 n < qr+1. Then the even number 2n can be represented as the sum of an
odd prime qi ∈ Qr and an odd number di = 2n − qi. (2) Based on the main assumption, all the odd
numbers di should be composite, then a group of linear algebraic equations can be formed and with
the solutions qi ∈ Qr. (3) When a contradiction between the expectation and the actual results of the
solutions is obtained, the main assumption is proved false so that the main theorem is proved.
It has been found that n 0 = 31, 637. Based on the main theorem and by verifying for n ≤ n 0 , it was
proved that for every number n greater than 1, there are always at least one pair of primes p and q
which are symmetrical about the number n so that even numbers greater than 2 can be expressed as
the sum of two primes. Hence, Goldbach’s conjecture was proved.
Also based on the main theorem, some theorems of the distribution of primes in short intervals were
10 Shan-Guang Tan
(3) In Section (3), by using the analysis of the matrix equation (2.8) with the theory of
linear algebra, solutions of the matrix equation (2.8) is derived by proving Lemma (3.1). Then
by proving lemmas (3.2-3.3) on solutions of the matrix equation, it can be proved that at least
one of the solutions is not equal to its corresponding expected solution qi ∈ Qr when n → ∞,
and there exists a finite positive number n 0 such that the actual solutions xi to the group of
linear algebraic equations are not equal to qi ∈ Qr when n > n 0. A contradiction.
(4) This contradiction shows that not all the odd numbers di are composite, so that the main
assumption (2.1) is false. Hence when n > n 0 , at least one odd number di is not a composite
but an odd prime greater than 2n −
2 n. Therefore the main theorem (4.1) is proved.
The assertion in the main theorem (4.1) has been verified for 2n = N ≤ 6. 5 × 10
10
. It has
been found that n 0 = 31, 637.
Based on the main theorem (4.1) for n > n 0 and by verifying for n ≤ n 0 , it was proved that
for every number n greater than 1, there are always at least one pair of primes p and q which
are symmetrical about the number n so that even numbers greater than 2 can be expressed as
the sum of two primes. Hence, Goldbach’s conjecture was proved.
1.2 Distribution of primes in short intervals
Based on the main theorem (4.1), some theorems of the distribution of primes in short intervals
were proved.
- It was proved that for every positive number m there is always one prime between m
2
and (m + 1) 2
. Thus the Legendre’s conjecture[10-11] was proved.
- It was proved that for every positive number m there are always two primes between
m 2 and (m + 1) 2
. One is between m 2 and m(m + 1), and another is between m(m + 1) and
(m + 1) 2
. Thus the Oppermann’s conjecture[19] was proved.
- It was proved that for every positive number m there are always three primes between
m
3 and (m + 1)
3
. The first one is between m
3 and m
2 (m + 1), and the second one is between
m 2 (m + 1) and m(m + 1) 2 , and the third one is between m(m + 1) 2 and (m + 1) 3
. So, the
theorem can be used to determine the Mills’ constant[17].
- It was proved that for every positive number m there are always k primes between m k
and (m + 1)
k where k is a positive integer greater than 1. For i = 1, 2 , · · · , k, there is always
one prime between m
k−i+ (m + 1)
i− 1 and m
k−i (m + 1)
i .
- It was proved that there are at least two primes between p 2 i and pipi+1, and there are
also at least two primes between pipi+1 and p
2 i+1, for^ i >^ 1, where^ pi^ is the^ ith^ prime. Thus the
Hanssner’s conjecture[8] was proved.
- It was proved that there are at least four primes between p 2 i and p 2 i+ , for i > 1, where
pi is the ith prime. Thus the Brocard’s conjecture[20] was proved.
- It was proved that for every positive number m there are always four primes between m
3
and (m + 1) 3
. Thus the theorem can be used to determine the Mills’ constant[17].
- It was proved that for every positive number m ≥ a
k − 1 where a and k are integers
greater than 1, if there are at least Sk primes between m
k and (m + 1)
k , then there are always
Sk+1 primes between m k+ and (m + 1) k+ where Sk+1 = aSk.
- It was proved that there is always a prime between m − m
θ and m where θ = 1/2 and
m is a positive number greater than a positive number 2n 0.
Representation of numbers and distribution of primes 11
- It was proved that there is always a prime between m and m + m θ where θ = 1/2 and
m is a positive number greater than a positive number 2n 0.
- It was proved that the inequality
pi+1 −
pi < 1 holds for all i > 0, where pi is the
ith prime. Thus the Andrica’s conjecture[18] was proved.
- It was proved that there is always one prime between km and (k + 1)m where m is a
positive number greater than 1 and 1 ≤ k < m.
- It was proved that if the numbers 1, 2 , 3 , · · · , m
2 with m > 1 are arranged in m rows
each containing m numbers, then each row contains at least one prime. Thus the Sierpinski’s
conjecture[8] was proved.
- It was proved that between any two triangular numbers, there is at least one prime.
Thus the Sierpinski’s conjecture of triangular numbers[8] was proved.
2 Definitions
2.1 Sets of primes
Let n denote a positive number greater than or equal to 2 3 , an even number N = 2n and P
denote a set of all odd primes smaller than or equal to the number n where
P = {p 2 , p 3 , · · · , pl}, 3 = p 2 < p 3 < · · · < pl ≤ n. (2.1)
To select primes q ∈ P not being the prime factor of the number n, let Q denote a subset
of P and m be the number of elements of Q where
Q = {q|q ∈ P, gcd(q, n) = 1}, 3 ≤ q 1 < q 2 < · · · < qm < n. (2.2)
To select primes q ∈ Q smaller than
2 n, let take the first r elements of Q to form a prime
set Qr, that is
Qr = {q 1 , q 2 , · · · , qr } ⊂ Q (2.3)
where for n > 15 the number r is determined by the inequalities
qr <
2 n < qr+1.
Hence the number r is dependent on the number n. Generally speaking, the number r is
increased when the number n is increased.
Let amin be a real and consider the following equation
aminqr + qr+1 = 2n.
Since (
2 n/4) 3
n for n > 2 9 , there are at most two primes which are divisors of the
number n in the region (qr , qr+1). Then by Chebyshev-Bertrand theorem the inequality qr+1 <
8 qr holds. Hence we have
amin =
2 n − qr+
qr
2 n − 8 qr
qr
2 n − 8. (2.4)
2.2 Linear algebraic equations
Let an assumption contradict Goldbach’s conjecture as following:
Definition 2.1 (Main assumption) At least one even number greater than 2 n 0 can not be
expressed as the sum of two odd primes where n 0 ≥ 2.
Representation of numbers and distribution of primes 13
where all diagonal elements ai,i for i = 1, 2 , · · · , r are equal to 1 and there is just one non-
diagonal and non-zero element ai,ji in each row of the matrix A. For i = 1, 2 , · · · , r, 1 ≤ ji ≤ r
and ji 6 = i, ai,ji is an odd number greater than amin and satisfies gcd(ai,ji , n) = 1, and qi ∈ Qr
is neither a divisor of ai,j i nor a divisor of n.
It is easy to prove that if |det(A)| = 1 then all solutions of Equation (2.8) will be composite.
Thus it contradicts the fact that all solutions of Equation (2.8) should be primes smaller than √
2 n. Therefore there should be |det(A)| 6 = 1. In fact, by the theory of linear algebra, |det(A)|
should be an even number.
Since |det(A)| ≥ 2, there is det(A) 6 = 0. Therefore let A
− 1 and B denote the inverse and
adjoint matrix of A, respectively.
Lemma 2.3 By exchanging its rows and columns, the matrix A in Definition (2.2) can only
be transformed into one of three forms of a matrix A¯ according to the value of |det(A)|.
The first form of
A is
A =
1 0 0 · · · 0 a 1 ,r¯
a 2 , 1 1 0 · · · 0 0
0 a 3 , 2 1 · · · 0 0
0 0 a 4 , 3
0 0 0 · · · a¯r,r¯− 1 1
where rank(
A) = ¯r with ¯r = r; all diagonal elements ai,i are equal to 1; and there is just one
non-diagonal and non-zero element ai,i− 1 in each row of the matrix A¯ with i − 1 ≡ ¯r, 1 , · · · , r¯ −
1(mod r¯) for i = 1, 2 , · · · , r¯.
The second form of A¯ is
A^ ¯ =
As 0 0 0 0
· · · as+1,js+1 · · · 1 0 0 0
· · · as+2,js+2 · · · · · · 1 0 0
. . .
· · · ar,j r
where As is a principal square sub-matrix of A with rank(As) = s and similar to Form (2.9);
all diagonal elements ai,i are equal to 1; and there is just one non-diagonal and non-zero element
ai,ji in each row of the matrix
A with ji < i for rows i = s + 1, s + 2, · · · , r.
The third form of A¯ is
A =
A 1 0 0 0
0 A 2 0 0
0 0 0 Am
14 Shan-Guang Tan
where any one of A 1 , A 2 , · · · , Am is a principal square sub-matrix of A and similar to Form
(2.9) or (2.10).
Moreover, the matrix A or at least one of its principal square sub-matrixes can be trans-
formed into a matrix A¯ in Form (2.9) with r¯ ≤ r.
Proof For the matrix A in Definition (2.2), since |det(A)| 6 = 1, according to Equality (2.7),
the matrix A can not be transformed into a triangular matrix.
If at least one column of the matrix A contains only one element, then the matrix A can
only be transformed into the second or third form of A¯.
Otherwise for each column of the matrix A contains just two elements, then the matrix A
can only be transformed into the first or third form of A¯.
These results above can also be proved by induction as follows.
For r = 2, the matrix A can only be of the first form of A¯, that is,
A =
1 a 1 , 2
a 2 , 1 1
For r = 3, the matrix A can only be of the first form of A¯, that is,
A =
1 0 a 1 , 3
a 2 , 1 1 0
0 a 3 , 2 1
where each column of the matrix A contains just two elements, or the second form of
A, that
is,
A =
1 a 1 , 2 0
a 2 , 1 1 0
0 a 3 , 2 1
or A =
1 a 1 , 2 0
a 2 , 1 1 0
a 3 , 1 0 1
where at least one column of the matrix A contains only one element, and the matrix A contains
a principal square sub-matrix As with rank(As) = s = 2 and similar to Form (2.9).
Assume that for r = k, the matrix A in Definition (2.2) can only be transformed into one
of the three forms of a matrix
A according to the value of |det(A)|. If at least one column
of the matrix A contains only one element, then the matrix A can only be transformed into
the second or third form of
A. Otherwise for each column of the matrix A contains just two
elements, then the matrix A can only be transformed into the first or third form of
A.
Then for r = k + 1, if at least one column of the matrix A contains only one element, then
by exchanging its rows and columns, the matrix A can be transformed into the following form,
that is,
A =
Ak 0
· · · ak+1,j · · · 1
where Ak is a principal square sub-matrix of A with rank(Ak) = k and 1 ≤ j ≤ k. By
the assumption above, the matrix Ak in Definition (2.2) can only be transformed into one of
16 Shan-Guang Tan
Proof Since A¯ B¯ = det( A¯)I where I is an identity matrix, there should be
ai,ibi,j + ai,i− 1 bi− 1 ,j = bi,j + ai− 1 bi− 1 ,j = {
det(
A), i = j,
0 , i 6 = j.
where i − 1 ≡ ¯r, 1 , · · · , ¯r − 1(mod ¯r) for i = 1, 2 , · · · , ¯r.
It can be proved that by using Expression (2.14), the above equalities hold.
According to Expression (2.14) for j = i, since i − 1 ≡ ¯r(mod r¯) for i = 1, there are
bi,j + ai− 1 bi− 1 ,j = bi,i + ai− 1 bi− 1 ,i
1 + a¯r ∗ (−1) ¯r+
∏r¯− 1
k= ak = det( A¯), i = 1,
1 + ai− 1 ∗ (−1) ¯r+
∏i− 2
k= ak
∏¯r
k=i ak = det( A¯), i = 2, 3 , · · · , ¯r.
According to Expression (2.14) for j = i − 1, since j ≡ ¯r(mod ¯r) for j = 0, there are
bi,j + ai− 1 bi− 1 ,j = bi,j + ai− 1 bi− 1 ,i− 1 = bi,j + ai− 1 = ai− 1
i+¯r+¯r
i− 1 k= ak
¯r k=¯r ak = a¯r − a¯r = 0, i = 1
i+i− 1
i− 1 k=i− 1 ak = ai− 1 − ai− 1 = 0, i = 2, 3 , · · · , r.¯
According to Expression (2.14) for 1 ≤ j < i − 1 < ¯r, there are
bi,j + ai− 1 bi− 1 ,j = bi,j + ai− 1 ∗ (−1)
i−1+j
i− 2 ∏
k=j
ak
= bi,j − (−1)
i+j
i∏− 1
k=j
ak = bi,j − bi,j = 0.
According to Expression (2.14) for 1 < j < i − 1 = ¯r, there are
bi,j + ai− 1 bi− 1 ,j = bi,j + a¯r ∗ (−1)
r¯+j
r ¯− 1 ∏
k=j
ak
= bi,j − (−1)
i+j+¯r
i− 1 ∏
k=
ak
¯r ∏
k=j
ak = bi,j − bi,j = 0
where since i ≡ 1(mod ¯r) for bi,j , there are bi,j = b 1 ,j and
∏i− 1
k= ak =
k= ak = 1.
According to Expression (2.14) for 1 < i < j = 3, 4 , · · · , r¯, there are
bi,j + ai− 1 bi− 1 ,j = bi,j + ai− 1 ∗ (−1)
i−1+j+¯r
i− 2 ∏
k=
ak
r ¯ ∏
k=j
ak
= bi,j − (−1)
i+j+¯r
i− 1 ∏
k=
ak
¯r ∏
k=j
ak = bi,j − bi,j = 0.
Thus the adjoint matrix of A¯ is B¯ with elements defined by Expression (2.14).
The proof of the lemma is completed.
Representation of numbers and distribution of primes 17
3 Lemmas of linear algebraic equations
3.1 Solutions of linear algebraic equations
Lemma 3.1 For a matrix
A in Form (2.9), solutions to Equation (2.8) are
xi =
2 n
1 + ai
∑r¯ j=1 ci−^1 ,j ∑¯r j=1 ci,j
, i = 1, 2 , · · · , ¯r (3.1)
where i − 1 ≡ ¯r, 1 , · · · , r¯ − 1(mod r¯) for i = 1, 2 , · · · , r¯, and
ci,i = aibi,i = ai, i = 1, 2 , · · · , r,¯
ci,j = aibi,j = (−1)
i+j
i k=j ak, j < i = 2, 3 , · · · , r,¯
ci,j = aibi,j = (−1)
i+j+¯r ∏i
k= ak
∏r¯
k=j ak, i < j = 2, 3 , · · · , r,¯
where a 1 , a 2 , · · · , a¯r are defined by Expression (2.12), and bi,j are elements of B¯ determined by
Lemma (2.4).
Proof According to Expression (2.14) and for ¯r + 1 ≡ 1(mod ¯r), we have
max{|bi,j ||j = 1, 2 , · · · , ¯r} = |bi,i+1| = a
− 1 i
¯r ∏
k=
ak, i = 1, 2 , · · · , ¯r
and
bi,j bi,i+
i+j+1 1 ∏j− 1 k=i+ ak
, i = 1, 2 , · · · , r¯ − 1 , j > i,
bi,j bi,i+
i+j+1 1 ∏i− 1 k=j+ ak
∏ ¯r k=i+ ak
, i = 1, 2 , · · · , r¯ − 1 , j ≤ i,
b¯r,j br,¯¯r+
b¯r,j b¯r, 1
j+1 1 ∏j− 1 k=1 ak
, j = 2, 3 , · · · , r.¯
Thus, solutions to Equation (2.8) should be
xi = 2n
∑r¯
j= bi,j
r¯+ a 1 a 2 · · · a¯r
, i = 1, 2 , · · · , r.¯ (3.3)
The ratio of xi to xk in Solution (3.3) is
xi
xk
¯r j= bi,j ∑ ¯r j= bk,j
, i, k = 1, 2 , · · · , ¯r. (3.4)
According to Expression (3.2) and for ¯r + 1 ≡ 1(mod ¯r), we have
max{|ci,j ||j = 1, 2 , · · · , r¯} = |ci,i+1| =
r ¯ ∏
k=
ak, i = 1, 2 , · · · , ¯r, (3.5)
ci,j ci,i+
i+j+1 1 ∏j− 1 k=i+1 ak
, i = 1, 2 , · · · , r¯ − 1 , j > i,
ci,j ci,i+
i+j+1 1 ∏i− 1 k=j+1 ak
∏ ¯r k=i+1 ak
, i = 1, 2 , · · · , r¯ − 1 , j ≤ i,
c¯r,j cr,¯¯r+
c¯r,j c¯r, 1
j+1 1 ∏j− 1 k=1 ak
, j = 2, 3 , · · · , r,¯
and ¯r ∑
j=
ci,j
ci,i+
ai+
ai+1ai+
¯r− 1
¯r− 1 ∏
j=
ai+j
where i = 1, 2 , · · · , ¯r and i + j ≡ 1 , 2 , · · · , ¯r(mod ¯r) for i + j = 1, 2 , · · ·.
Representation of numbers and distribution of primes 19
on the other. Also there are
1 − μi+2(m)
ai+
¯r ∑
j=
ci,j
ci,i+
ai+
1 − νi+3(m ′ )
ai+1ai+
where
μi(0) = {
0 for ¯r is even
1 /ai for r¯ is odd
and for 1 ≤ ξ ≤ m = [(¯r − 2)/2]
μi(ξ) = (1 −
ai+
ai
μi+2(ξ − 1)
aiai+
ξ ∑
j=
ai+2j− 1
i+2j− 2 ∏
k=i
ak
0 for r¯ is even
∏i+2ξ
k=i
1 ak
for ¯r is odd
and
νi(0) = {
0 for ¯r is odd
1 /ai for r¯ is even
and for 1 ≤ ξ ≤ m
′ = [(¯r − 3)/2]
νi(ξ) = (1 −
ai+
ai
νi+2(ξ − 1)
aiai+
ξ ∑
j=
ai+2j− 1
i+2j− 2 ∏
k=i
ak
0 for ¯r is odd
∏i+2ξ
k=i
1 ak
for r¯ is even
These equalities are also obvious if we write
r ¯ ∑
j=
ci,j
ci,i+
ai+
ai+
ai+
3 ∏
j=
ai+j
ai+
ai+
m ′ ∑
j=
ai+2j+
i+2j+ ∏
k=i+
ak
0 for ¯r is odd
∏ i+2m ′
k=i+
1 ak for r¯ is even
1 − νi+3(m
′ )
ai+
ai+
on the one hand, and
¯r ∑
j=
ci,j
ci,i+
ai+
ai+
2 ∏
j=
ai+j
ai+
m ∑
j=
ai+2j+
i+2j ∏
k=i+
ak
0 for r¯ is even
∏i+2m+
k=i+
1 ak
for ¯r is odd
1 − μi+2(m)
ai+
on the other. Since
∑¯r
j= ci− 1 ,j ∑ ¯r j= ci,j
ci− 1 ,i
ci,i+
∑r¯
j=
ci− 1 ,j ci− 1 ,i ∑ ¯r j=
ci,j ci,i+
∑¯r
j=
ci− 1 ,j ci− 1 ,i ∑ ¯r j=
ci,j ci,i+
20 Shan-Guang Tan
for i = 1, 2 , · · · , r¯, we obtain
ai
r ¯ ∑
j=
ci− 1 ,j
ci− 1 ,i
r ¯ ∑
j=
ci,j
ci,i+
1 − 1 /ai + (1 − νi+2(m
′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
1 + (μi+2(m) − νi+2(m
′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
ai
where 1 − 1 /ai + (1 − νi+2(m ′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
1 + (1 − νi+2(m ′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
ai
1 − μi+2(m)
aiai+
1 − (1 − μi+2(m))/ai+
1 + (1 − νi+2(m
′ ))/(aiai+1) − (1 − μi+2(m))/(aiai+1)
1 − (1 − μi+2(m))/ai+
ai
1 + (μi+2(m) − νi+2(m
′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
ai
that is
ai
¯r ∑
j=
ci− 1 ,j )/(
r ¯ ∑
j=
ci,j ) =
1 + (μi+2(m) − νi+2(m
′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
ai
Thus, we have
ai < 1 + ai(
¯r ∑
j=
ci− 1 ,j )/(
¯r ∑
j=
ci,j ) = ai
1 + (μi+2(m) − νi+2(m
′ ))/(aiai+1)
1 − (1 − μi+2(m))/ai+
and
1 − (1 − μi+2(m))/ai+
1 + (μi+2(m) − νi+2(m ′ ))/(aiai+1)
ai
1 + ai(
¯r j= ci− 1 ,j )/(
¯r j= ci,j )
so that according to Expression (3.1) there are
xi
2 n/ai
ai
1 + ai(
∑r¯
j= ci− 1 ,j )/(
∑r¯
j= ci,j )
and since 1 − t <
1 1+t for 0 < |t| < 1
1 − μi+2(m)
ai+
μi+2(m) − νi+2(m ′ )
aiai+
xi
2 n/ai
Now by Equality (2.7) with a matrix A¯ in Form (2.9), there are
2 n
ai
= qi +
qi+
ai
for i = 1, 2 , · · · , r¯
where ¯r + 1 ≡ 1(mod ¯r).
Thus, we have
2 n
ai
1 − μi+2(m)
ai+
μi+2(m) − νi+2(m
′ )
aiai+
2 n
ai
[1 −
1 − μi+2(m)
ai+
μi(m + 1)(μi+2(m) − νi+2(m
′ ))
ai+
]
2 n
ai
2 n
ai+
1 − μi+2(m)
ai
2 nμi(m + 1)(μi+2(m) − νi+2(m ′ ))
aiai+
