# There are infinitely many sets of N-odd prime numbers and pairs of consecutive odd prime numbers.

Let us consider positive odd numbers which share a prime
factor>1 as a kind, then the positive directional half line of the
number axis consists of infinite many equivalent line segments on same
permutation of X kinds' odd points plus odd points amongst the X
kinds' odd points, where X [greater than or equal to] 1. We will
prove together that there are infinitely many sets of n-odd prime
numbers and pairs of consecutive odd prime numbers by the mathematical
induction with aid of such equivalent line segments and odd points
thereof, in this article.

Keywords Sets of n-odd prime numbers, Pairs of consecutive odd prime numbers, Mathematical induction, Odd points, Positive directional half line of the number axis, [RLSS.sub.No1~NoX], Sets of *[mu](*s)+b([??]s)*, Pairs of *v([??]s)*, The coexisting theorem, No1 [RLS.sub.No1~NoX], Set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Basic Concepts

Suppose n >1, and [[kappa].sub.1] < [[kappa].sub.2] < ... < [[kappa].sub.n-1] are n-1 natural numbers, and [J.sub.x], [J.sub.x] + [[kappa].sub.1], [J.sub.x] + [[kappa].sub.2], [J.sub.x] + [[kappa].sub.3], ... [J.sub.x] + [[kappa].sub.n-1] are all odd prime numbers, then we call ([J.sub.x],[J.sub.x] + [[kappa].sub.1], [J.sub.x] + [[kappa].sub.2], Jx + [[kappa].sub.3],...[J.sub.x] + [[kappa].sub.n-1]) a set of n-odd prime numbers. Thereupon we conjecture that for any positive odd prime number [J.sub.p], if a number of residue's classes which n integers 0, [[kappa].sub.1], ... [[kappa].sub.n-1] divide respectively by modulus [J.sub.p] is less than [J.sub.p], then there are infinitely many sets of n-odd prime numbers which differ orderly by [[kappa].sub.1], [[kappa].sub.2]- [[kappa].sub.1], [[kappa].sub.3]- [[kappa].sub.1],...and [[kappa].sub.n-1]- [[kappa].sub.n-2]. We term the conjecture as n-odd prime numbers' conjecture. For example, when n [greater than or equal to] 2, and [[kappa].sub.1]=2, it contains twin prime numbers' conjecture. Also, when n [greater than or equal to] 3, [[kappa].sub.1]=2 and [[kappa].sub.1]=6, it contains 3-odd prime numbers' conjecture. And so on and so forth ...

Evidently, if modulus [J.sub.p] [greater than or equal to] [J.sub.x] + [[kappa].sub.n-1], then each odd prime number of such a set of odd numbers belongs in a residue class, thus number n of n-odd prime numbers is less than [J.sub.p]. If modulus [J.sub.p] < [J.sub.x], then number n of n-odd prime numbers may be greater than [J.sub.p]. For example, a set of 16-odd prime numbers (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73) for modulus [J.sub.4] (i.e. 11), it has 16 odd prime numbers of 10 residue's classes because 17[equivalent to]61(mod 11), 19[equivalent to]41(mod 11), 23[equivalent to]67(mod 11), 29[equivalent to]73 (mod 11), 31[equivalent to]53(mod 11), and 37[equivalent to]59(mod 11) plus 13, 43, 47, and 71.

In addition, there is such a conjecture, namely if there is a pair of consecutive odd prime numbers which differ by 2k, then there are surely infinitely many pairs of consecutive odd prime numbers which differ by 2k, where k is a natural number. This conjecture needs still us to prove it. When k=1, it is the very twin prime numbers' conjecture evidently.

Everyone knows, each and every odd point at positive directional half line of the number axis expresses a positive odd number. Also infinite many a distance between two consecutive odd points at the positive directional half line equal one another.

Let us use the symbol "*" to denote an odd point, whether * is in a formulation or it is at the initial positive directional half line of the number axis. Moreover the positive directional half line is marked merely with symbols of odd points. Please, see following first illustration.

[ILLUSTRATION OMITTED]

We use also symbol "*s" to denote at least two odd points in formulations.

Then, the number axis's positive directional half line which begins with odd point 3 is called the half line for short thereinafter.

We consider smallest positive odd prime number 3 as No1 odd prime number, and consider positive odd prime number [J.sub.x] as [No.sub.X] odd prime number, where X [greater than or equal to] 1, then odd prime number 3 is written as [J.sub.1] as well.

And then, we consider positive odd numbers which share prime factor [J.sub.x] as [No.sub.X] kind of odd numbers. If an odd number contains [alpha] one another's-different prime factors, then the odd number belongs in the [alpha] kinds of odd numbers concurrently, where [alpha] [greater than or equal to] 1.

There is an unique odd prime number [J.sub.x] within [No.sub.X] kind's odd numbers, yet we term others as [No.sub.X] kind of odd composite numbers.

If one * is defined as an odd composite point, then we must change symbol "[??]" for its symbol "*". And use symbol "[??]s" to denote at least two definite odd composite points in formulations.

In course of the proof, we shall change [??]s for * s at places of [SIGMA] [No.sub.X] [X [greater than or equal to] 1] kind's odd composite points according as X is from small to large.

Since [No.sub.X] kind's odd numbers are infinitely many a product which multiplies every odd number by [J.sub.x], so there is a [No.sub.X] kind's odd point within consecutive [J.sub.x] odd points at the half line.

Therefore any one another's permutation of X kind's odd points plus odd points amongst the X kind's odd points assumes always infinite many recurrences on same pattern at the half line, irrespective of their prime/composite attribute.

We analyze seriatim [No.sub.X] kind of odd points at the half line according to X =1, 2, 3 ... in one by one, and range them as second illustration.

[ILLUSTRATION OMITTED]

We consider one another's equivalent shortest line segments at the half line in accordance with same permutation of X kinds' odd points plus odd points amongst the X kinds' odd points as recurring segments of the X kinds' odd points.

We use character "[RLS.sub.No1~NoX]" to express a recurring segment of [SIGMA] [No.sub.X] [X [greater than or equal to]1] kind of odd points, and use character "[RLSS.sub.No1~NoX]" to express the plural. If one * is affirmed as an odd prime point, then this * is rewritten as one [??] at the half line and/or in formulations, and symbol [??]s express at least two odd prime points in formulations. For example, the one another's permutation of certain kinds of odd points at No1 [RLS.sub.No1~No4,] please, see following third illustration.

The permutation of odd prime & composite points at No1 [RLS.sub.No1~No4]

[ILLUSTRATION OMITTED]

Annotation: "[??]" denotes an odd prime point; "[??]" denotes an odd composite point. No1 [RLS.sub.No1] ends with odd point 7; No1 [RLS.sub.No1~No2] ends with odd point 31; No1 [RLS.sub.No1~No3] ends with odd point 211; No1 [RLS.sub.No1~No4] ends with odd point 2311.

Justly No1 [RLS.sub.No1~NoX] begins with odd point 3. There are [PI][J.sub.x] odd points at each [RLS.sub.No1~NoX], where X [greater than or equal to] 1, and [PI][J.sub.x] = [J.sub.1]*[J.sub.2]* ... *[J.sub.x].

Undoubtedly one [RLS.sub.No1~No(X+1)] consists of consecutive [J.sub.x+1] [RLSS.sub.No1~NoX], and they link one by one.

Since none of any kind's odd composite points coincides with odd point 1 on the left of No1 [RLS.sub.No1~NoX], then none of any kind's odd composite points coincides with the odd point which closes on the left of No2 [RLS.sub.No1~NoX] according to the definition of recurring segments of the X kinds' odd points. The odd point which closes on the left of No2 [RLS.sub.No1~NoX] is exactly the most right odd point of No1 [RLS.sub.No1~NoX]. Thus the most right odd point of No1 [RLS.sub.No1~NoX] is an odd prime point always. Namely 2[PI][J.sub.x]+1 is an odd prime number always.

Number the ordinals of odd points at seriate each [RLS.sub.No1~NoX+y] by consecutive natural numbers which begin with 1, namely from left to right each odd point at seriate each [RLS.sub.No1~NoX+y] is marked with from small to great a natural number [greater than or equal to]1 in the proper order, where y [greater than or equal to]0.

Then, there is one No(X+y) kind's odd point within [J.sub.x+y] odd points which share an ordinal at [J.sub.x+y] [RLSS.sub.No1~No(x+y-1)] of a [RLS.sub.No1~Nox+y].

Furthermore, there is one No(X+y) kind's odd composite point within [J.sub.x+y] odd points which share an ordinal at [J.sub.x+y] [RLSS.sub.No1~No(x+y-1)] of seriate each [RLS.sub.No1~Nox+y] on the right of No1 [RLS.sub.No1~Nox+y].

Odd prime points [J.sub.1], [J.sub.2] ... [J.sub.x-1] and [J.sub.x] are at No1 [RLS.sub.No1~NoX]. Yet, there are X odd composite points on ordinals of [J.sub.1] plus [J.sub.2] ... plus [J.sub.x-1] plus [J.sub.x] at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX]. Thus No1 [RLS.sub.No1~NoX] is a particular [RLS.sub.No1~NoX] in contradistinction to each of others.

After change [??]s for * s at places of [SIGMA]NoX [X[greater than or equal to] 1] kind's odd composite points at the half line, if one * is separated from another * by [mu] *s plus b [??]s irrespective of their permutation, then express such a combinative form as a set of * [mu](*s)+b([??]s) *, where [mu] [greater than or equal to] 0, and b [greater than or equal to] 0.

If [mu]+2 *s of * [mu](*s)+b([??]s) * are all defined as odd prime points, then the set of * [mu](*s)+b([??]s) * is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Further, if the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] lies within consecutive [J.sub.x] odd points, and for odd prime number [J.sub.x], a number of residue's classes which [mu]+2 odd prime numbers whereof [mu]+2 [??]s express divided respectively by modulus [J.sub.x] is less than [J.sub.x], then, such a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the very a set of n-odd prime points, where n=[mu]+2.

If two *s of * v([??]s) * are defined as odd prime points, then the pair of *v([??]s)* is rewritten as a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where v [greater than or equal to] 0.

When [mu]=0, a set of * [mu](*s)+b([??]s) * is exactly a pair of *b([??]s)*, and a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is exactly a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where b [greater than or equal to] 0.

Let [mu]+b=m, a set of * [mu](*s)+b([??]s) * may be written as a set of * m(*s[??]s) *, and a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] may be written as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

After change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, [J.sub.x-h] at No1 [RLS.sub.No1~NoX] is defined as an odd prime point, where X>h [greater than or equal to] 0, yet there are infinitely many *s on the right of [J.sub.x] at the half line, and every * is an undefined odd point on prime/composite attribute. Anyhow every prime factor of an odd number which each * at the right of [J.sub.x] expresses is greater than [J.sub.x].

A set of * [mu](*s)+b([??]s) * is negated according as any * of the set is defined as one Also a pair of * v([??]s) * is negated according as either * of the pair is defined as one If a set of * [mu](*s)+b([??]s) * can not always be negated, then it is precisely a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Likewise, if a pair of * v([??]s) * can not always be negated, then it is precisely a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the definition for recurring segments of X kinds' odd points, we can conclude that after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, if there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.x] odd points on the right of [J.sub.x] at No1 [RLS.sub.No1~NoX], then there is surely a set of * [mu](*s)+b([??]s) * on ordinals of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX].

Without doubt, the converse proposition is tenable too. Namely after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, if there is a set of * [mu](*s)+b([??]s)* within consecutive [J.sub.x] odd points at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX], and from left to right Nok odd prime points of all sets of *[mu](*s)+b([??]s) * share an ordinal, then there is surely a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ordinals of any such set of * [mu](*s)+b([??]s) *, at No1 [RLS.sub.No1~NoX], where k = 1, 2, ... [mu]+2.

Of course, every [??] of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and every prime factor of an odd number which each * of every such set of * [mu](*s)+b([??]s) * expresses are greater than [J.sub.x].

To be brief, after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.x] at No1 [RLS.sub.No1~NoX] and infinite many sets of * [mu](*s)+b([??]s)* on ordinals of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate [RLSS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX] coexist at the half line.

We term the aforesaid conclusion as the coexisting theorem of a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus infinite many sets of * [mu](*s)+b([??]s)* at the half line, or term it as the coexisting theorem for short.

[J.sub.x+1] [RLSS.sub.No1~NoX] of any [RLS.sub.No1~No(x+1)] may be folded at an illustration, one by one, so as to view conveniently, e.g. No1, No2 and No3 kinds' odd points at two [RLSS.sub.No1~No3] from the differentia, please, see following fourth illustration.

[ILLUSTRATION OMITTED]

After change [??]s for *s at places of No1 plus No2 plus No3 kinds' odd composite points, every [??] denotes a definite odd prime point, and every * denotes an undefined odd point at prime/composite attribute, and every [??] denotes a definite odd composite point, in the illustration. Line segment 3(211) is No1 [RLS.sub.No1~No3] and line segment CD is any of seriate [RLSS.sub.No1~No3] on the right of No1 [RLS.sub.No1~No3].

The Proof

We will prove together that there are infinitely many sets of n-odd prime numbers and pairs of consecutive odd prime numbers by the mathematical induction with the aid of [RLSS.sub.No1~NoX] and odd points thereof, thereinafter.

1. When X=1, there is a set of [??][??] alone on the right of [J.sub.1] at No1 [RLS.sub.No1], and the set of [??][??] is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as well, i.e. twin odd prime points 5 and 7, where [v.sub.1]=0.

When X=2, there are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.2] at No1 [RLS.sub.No1~No2], and these odd points contain several sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.s] odd points, including several pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within them, where [[mu].sub.2] [less than or equal to] 6, [b.sub.2] [less than or equal to] 5, [J.sub.1] [less than or equal to] [J.sub.s] [less than or equal to] [J.sub.5], and [v.sub.2] [less than or equal to] 2.

Evidently these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] contain pairs of twin odd prime points.

When X=3, there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.f] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.3] at No1 [RLS.sub.No1~No3], where [[mu].sub.2] [less than or equal to] [[mu].sub.3] [less than or equal to] 41, [b.sub.2] [less than or equal to] [b.sub.3] [less than or equal to] 58, [J.sub.s] [less than or equal to] [J.sub.f] [less than or equal to] [J.sub.27] =101, and [v.sub.2] [less than or equal to] [v.sub.3] =0, 1, 2, 3, 4, 5 and 6.

Evidently these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody certain sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody all pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on values of [v.sub.3] at No1 [RLS.sub.No1~No3], we instance (11, 13), (13, 17), (23, 29), (89, 97), (139, 149), (199, 211) and (113, 127). Please, see preceding third illustration once again.

When X=4, there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.a] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.4] at No1 [RLS.sub.No1~No4], where [[mu].sub.3] [less than or equal to] [[mu].sub.4] [less than or equal to] 337, [b.sub.3] [less than or equal to] [b.sub.4] < 813, [J.sub.f] [less than or equal to] [J.sub.a] [less than or equal to] [J.sub.189] =1151, and [v.sub.3] [less than or equal to] [v.sub.4] =0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 16.

Evidently these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody certain sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody all pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on values of [v.sub.4] at No1 [RLS.sub.No1~No4], we instance (17, 19), (19, 23), (31, 37), (89, 97), (139, 149), (211, 223), (293, 307), (1831, 1847), (1259, 1277), (887, 907), (1669, 1693), (2179, 2203) and (1327, 1461). Please, see preceding third illustration once more.

2. When X=[beta][greater than or equal to]4, suppose that there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.b] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[beta]] at No1 [RLS.sub.No1~No[beta]], where [[mu].sub.[beta]] [greater than or equal to] [[mu].sub.4,], [b.sub.[beta]] [greater than or equal to][b.sub.4], v[beta] [greater than or equal to] v4, [J.sub.b] [greater than or equal to] [J.sub.a], and [J.sub.[beta]] [greater than or equal to] [J.sub.4]. In addition, these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody any of sets of n-odd prime points on the right of [J.sub.1] at No1 [RLS.sub.No1~No[psi]], and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody any of pairs of consecutive odd prime points at No1 [RLS.sub.No1~No[psi]], where [psi]<[beta].

Let us suppose that any of sets of n-odd prime points on the right of [J.sub.1] at No1 [RLS.sub.No1~No[psi]] is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ; and any of pairs of consecutive odd prime points at No1 [RLS.sub.No1~No[psi]] is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where [[mu].sub.p] [greater than or equal to] [[mu].sub.4], [b.sub.q] [greater than or equal to] [b.sub.4], and [v.sub.[delta]] [greater than or equal to] [v.sub.4].

3. When X=[eta]>[beta], prove that there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.c] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[eta]] at No1 [RLS.sub.No1~No[eta]], where [[mu].sub.[eta]] [greater than or equal to] [[mu].sub.[beta]], [b.sub.[eta]] [greater than or equal to] b[beta], v[eta] [greater than or equal to] [v.sub.[beta]] [J.sub.c] [greater than or equal to] [J.sub.b], and [J.sub.[eta]] > [J.sub.[beta]]. In addition, these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must embody a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove, and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must embody a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove.

Proof . Since there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.b] odd points on the right of [J.sub.[beta]] at No1 [RLS.sub.No1~No[beta]], furthermore, we name the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]", where d [greater than or equal to] 1 and g =[beta]+d+[[mu].sub.p]+1. Well then, let us first prove that there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ordinals of [J.sub.[beta]]+d [[mu].sub.p]([??]s)+bq([??]s) [J.sub.g] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog], hereinafter.

We know that every odd number >1 has a smallest prime factor except for 1 surely, yet the smallest prime factor of any odd prime number is exactly it itself.

If greatest one within respective smallest prime factors of [b.sub.q] odd composite numbers whereof [b.sub.q]([??]s) between [J.sub.[beta]+d] and [J.sub.g] express is written as [J.sub.[phi]], then the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is either at No1 [RLS.sub.No1~No[phi]] or out of No1 [RLS.sub.No1~No[phi]]. If it is at No1 [RLS.sub.No1~No[phi]], then let 1 [less than or equal to] X1 [less than or equal to] [phi]. If it is out of No1 [RLS.sub.No1~No[phi]], then suppose that it is just at No1 [RLS.sub.No1~No[kappa]], but it is not at No1 [RLS.sub.No1~No[kappa]-1] then [kappa] > [phi], and let 1[less than or equal to] X2 [less than or equal to][kappa].

If the set of [J.sub.[beta]+d] [[mu].sub.p] ([??]s)+[b.sub.q]([??]s) [J.sub.g] is at No1 [RLS.sub.No1~No[phi]] then after change [??]s for *s at places of [SIGMA][No.sub.X1][1 [less than or equal to] X1 [less than or equal to] [phi]] kind's odd composite points, there is a set of *[[mu].sub.p](*s)+[b.sub.q]([??]s)* on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] at seriate each [RLS.sub.No1~No[phi]] on the right of No1 [RLS.sub.No1~No[phi]].

If the set of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] is just barely at No1 [RLS.sub.No1~No[kappa]], but it is out of No1 [RLS.sub.No1~No([kappa-1]], then after change [??]s for *s at places of [SIGMA][No.sub.X2] [1 [less than or equal to] X2 [less than or equal to] [kappa]] kind's odd composite points, there is a set of * [[mu].sub.p](*s)+[b.sub.q]([??]s)* on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] at seriate each [RLS.sub.No1~No[kappa]] on the right of No1 [RLS.sub.No1~No[kappa]].

Either [J.sub.[phi]] or [J.sub.[kappa]] [greater than or equal to] [[mu].sub.p] + [b.sub.q] +2, uniformly let it to equal [J.sub.v]. If [J.sub.[phi]] or [J.sub.[kappa]] < [[mu].sub.p] + [b.sub.q] +2, then suppose that [J.sub.v] is the smallest odd prime number which is not smaller than [[mu].sub.p] + [b.sub.q] +2.

Each set of *[[mu].sub.p] (*s)+[b.sub.q]([??]s) * on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] considering aforementioned either case is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If some set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , then the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let v+1 [less than or equal to][omega][less than kor equal to]g, since there is one No[omega] kind's odd point within consecutive [J.sub.[omega]] odd points, and there is one No[omega] kind's odd point within [J.sub.[omega]] odd points which share an ordinal at seriate [J.sub.[omega]] [RLSS.sub.No1~No[omega]-1], therefore there is a series of results as the following.

After successively change [??]s for *s at places of No (v+1) kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p]) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p] sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No~No(v+1)] on the right of No1 [RLS.sub.No1~No(v+1)]. Of course, every prime factor of an odd number which each [??] at here expresses is greater than [J.sub.v+1].

After successively change [??]s for *s at places of No (v+2) kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p]) ([J.sub.v+2] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p]([J.sub.v+2] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~No(v+1)] on the right of No1 [RLS.sub.No1~No(v+1)]. Of course, every prime factor of an odd number which each [??] at here expresses is greater than [J.sub.v+2].

And so on and so forth ...

Up to after successively change [??]s for *s at places of Nog kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p])([J.sub.v+2] - 1)([J.sub.v+3] - 1)...([J.sub.g] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p]([J.sub.v+2] - 1)([J.sub.v+3] - 1)...([J.sub.g] - 1) pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~Nog] on the right of No1 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Of course, every prime factor of an odd number which each [??] at here expresses is greater than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the half line on the right of No1 [RLS.sub.No1~Nog] has infinitely many [RLSS.sub.No1~Nog], thus there are both infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and infinitely many pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the half line after successively change [??]s for *s at places of [SIGMA]No[omega] [v+1 [less than or equal to] [omega] [less than or equal to] g] kind's odd composite points. Concurrently, there are infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ) infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ... and infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the half line. Of course, every prime factor of an odd number which each [??] within aforementioned sundry sets expresses is greater than [J.sub.g].

Thus there are a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , ... a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog] according to aforesaid that coexisting theorem.

Thus it can seen, preceding results contain such a conclusion, namely there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog]. This is just the proposition which need us to prove.

In case a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove is embodied within the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... plus the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus the pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog], the pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is proven synchronously into the real too.

If a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove is not embodied within aforementioned sundry sets of n-odd prime points, then we can likewise apply the aforesaid way of doing according to the coexisting theorem to prove and get that there is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which embodies such a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog,] but values of g on two places are perhaps unlike.

Since the mathematical induction sets up a claim to X=[eta]>[beta], whereas now has X=g=[beta]+d+[[mu].sub.p]+1> [beta], thus can replace g by [eta], therefore, we have proven that there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[eta]] at No1 [RLS.sub.No1~No[eta]].

When vest further x with a value which is greater than g, we likewise can continue to apply the aforesaid way of doing and the coexisting theorem to prove and get that thre are another set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and another pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . And so on and so forth ...

Though values of X are not consecutive natural numbers under the prerequisite that it is proven there are sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the aforesaid way of doing and the coexisting theorem, but, since there are infinitely many natural numbers at all events, so that there are infinitely many values of X which accord with the claim. Therefore there are both infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and infinitely many pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expresses a set of n-odd prime numbers, where n=[[mu].sub.p]+2, consequently there are infinitely many sets of n-odd prime numbers.

In addition, a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expresses a pair of consecutive odd prime numbers which differ by 2([v.sub.[delta]]+1), consequently there are infinitely many pairs of consecutive odd prime numbers which differ by 2k, where k=[v.sub.[delta]]+1.

Taken one with another, we have proven that there are both infinitely many sets of n-odd prime numbers and infinitely many pairs of consecutive odd prime numbers which differ by 2k.

Zhang Tianshu

Nanhai west oil corporation, China offshore oil, Zhanjiang city, Guangdong province, P.R. China.

Email: (1) tianshu_zhang507@yahoo. com. cn;

(2) friend_zhang888@sina.com; (3) xinshijizhang@hotmail.com

Keywords Sets of n-odd prime numbers, Pairs of consecutive odd prime numbers, Mathematical induction, Odd points, Positive directional half line of the number axis, [RLSS.sub.No1~NoX], Sets of *[mu](*s)+b([??]s)*, Pairs of *v([??]s)*, The coexisting theorem, No1 [RLS.sub.No1~NoX], Set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Basic Concepts

Suppose n >1, and [[kappa].sub.1] < [[kappa].sub.2] < ... < [[kappa].sub.n-1] are n-1 natural numbers, and [J.sub.x], [J.sub.x] + [[kappa].sub.1], [J.sub.x] + [[kappa].sub.2], [J.sub.x] + [[kappa].sub.3], ... [J.sub.x] + [[kappa].sub.n-1] are all odd prime numbers, then we call ([J.sub.x],[J.sub.x] + [[kappa].sub.1], [J.sub.x] + [[kappa].sub.2], Jx + [[kappa].sub.3],...[J.sub.x] + [[kappa].sub.n-1]) a set of n-odd prime numbers. Thereupon we conjecture that for any positive odd prime number [J.sub.p], if a number of residue's classes which n integers 0, [[kappa].sub.1], ... [[kappa].sub.n-1] divide respectively by modulus [J.sub.p] is less than [J.sub.p], then there are infinitely many sets of n-odd prime numbers which differ orderly by [[kappa].sub.1], [[kappa].sub.2]- [[kappa].sub.1], [[kappa].sub.3]- [[kappa].sub.1],...and [[kappa].sub.n-1]- [[kappa].sub.n-2]. We term the conjecture as n-odd prime numbers' conjecture. For example, when n [greater than or equal to] 2, and [[kappa].sub.1]=2, it contains twin prime numbers' conjecture. Also, when n [greater than or equal to] 3, [[kappa].sub.1]=2 and [[kappa].sub.1]=6, it contains 3-odd prime numbers' conjecture. And so on and so forth ...

Evidently, if modulus [J.sub.p] [greater than or equal to] [J.sub.x] + [[kappa].sub.n-1], then each odd prime number of such a set of odd numbers belongs in a residue class, thus number n of n-odd prime numbers is less than [J.sub.p]. If modulus [J.sub.p] < [J.sub.x], then number n of n-odd prime numbers may be greater than [J.sub.p]. For example, a set of 16-odd prime numbers (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73) for modulus [J.sub.4] (i.e. 11), it has 16 odd prime numbers of 10 residue's classes because 17[equivalent to]61(mod 11), 19[equivalent to]41(mod 11), 23[equivalent to]67(mod 11), 29[equivalent to]73 (mod 11), 31[equivalent to]53(mod 11), and 37[equivalent to]59(mod 11) plus 13, 43, 47, and 71.

In addition, there is such a conjecture, namely if there is a pair of consecutive odd prime numbers which differ by 2k, then there are surely infinitely many pairs of consecutive odd prime numbers which differ by 2k, where k is a natural number. This conjecture needs still us to prove it. When k=1, it is the very twin prime numbers' conjecture evidently.

Everyone knows, each and every odd point at positive directional half line of the number axis expresses a positive odd number. Also infinite many a distance between two consecutive odd points at the positive directional half line equal one another.

Let us use the symbol "*" to denote an odd point, whether * is in a formulation or it is at the initial positive directional half line of the number axis. Moreover the positive directional half line is marked merely with symbols of odd points. Please, see following first illustration.

[ILLUSTRATION OMITTED]

We use also symbol "*s" to denote at least two odd points in formulations.

Then, the number axis's positive directional half line which begins with odd point 3 is called the half line for short thereinafter.

We consider smallest positive odd prime number 3 as No1 odd prime number, and consider positive odd prime number [J.sub.x] as [No.sub.X] odd prime number, where X [greater than or equal to] 1, then odd prime number 3 is written as [J.sub.1] as well.

And then, we consider positive odd numbers which share prime factor [J.sub.x] as [No.sub.X] kind of odd numbers. If an odd number contains [alpha] one another's-different prime factors, then the odd number belongs in the [alpha] kinds of odd numbers concurrently, where [alpha] [greater than or equal to] 1.

There is an unique odd prime number [J.sub.x] within [No.sub.X] kind's odd numbers, yet we term others as [No.sub.X] kind of odd composite numbers.

If one * is defined as an odd composite point, then we must change symbol "[??]" for its symbol "*". And use symbol "[??]s" to denote at least two definite odd composite points in formulations.

In course of the proof, we shall change [??]s for * s at places of [SIGMA] [No.sub.X] [X [greater than or equal to] 1] kind's odd composite points according as X is from small to large.

Since [No.sub.X] kind's odd numbers are infinitely many a product which multiplies every odd number by [J.sub.x], so there is a [No.sub.X] kind's odd point within consecutive [J.sub.x] odd points at the half line.

Therefore any one another's permutation of X kind's odd points plus odd points amongst the X kind's odd points assumes always infinite many recurrences on same pattern at the half line, irrespective of their prime/composite attribute.

We analyze seriatim [No.sub.X] kind of odd points at the half line according to X =1, 2, 3 ... in one by one, and range them as second illustration.

[ILLUSTRATION OMITTED]

We consider one another's equivalent shortest line segments at the half line in accordance with same permutation of X kinds' odd points plus odd points amongst the X kinds' odd points as recurring segments of the X kinds' odd points.

We use character "[RLS.sub.No1~NoX]" to express a recurring segment of [SIGMA] [No.sub.X] [X [greater than or equal to]1] kind of odd points, and use character "[RLSS.sub.No1~NoX]" to express the plural. If one * is affirmed as an odd prime point, then this * is rewritten as one [??] at the half line and/or in formulations, and symbol [??]s express at least two odd prime points in formulations. For example, the one another's permutation of certain kinds of odd points at No1 [RLS.sub.No1~No4,] please, see following third illustration.

The permutation of odd prime & composite points at No1 [RLS.sub.No1~No4]

[ILLUSTRATION OMITTED]

Annotation: "[??]" denotes an odd prime point; "[??]" denotes an odd composite point. No1 [RLS.sub.No1] ends with odd point 7; No1 [RLS.sub.No1~No2] ends with odd point 31; No1 [RLS.sub.No1~No3] ends with odd point 211; No1 [RLS.sub.No1~No4] ends with odd point 2311.

Justly No1 [RLS.sub.No1~NoX] begins with odd point 3. There are [PI][J.sub.x] odd points at each [RLS.sub.No1~NoX], where X [greater than or equal to] 1, and [PI][J.sub.x] = [J.sub.1]*[J.sub.2]* ... *[J.sub.x].

Undoubtedly one [RLS.sub.No1~No(X+1)] consists of consecutive [J.sub.x+1] [RLSS.sub.No1~NoX], and they link one by one.

Since none of any kind's odd composite points coincides with odd point 1 on the left of No1 [RLS.sub.No1~NoX], then none of any kind's odd composite points coincides with the odd point which closes on the left of No2 [RLS.sub.No1~NoX] according to the definition of recurring segments of the X kinds' odd points. The odd point which closes on the left of No2 [RLS.sub.No1~NoX] is exactly the most right odd point of No1 [RLS.sub.No1~NoX]. Thus the most right odd point of No1 [RLS.sub.No1~NoX] is an odd prime point always. Namely 2[PI][J.sub.x]+1 is an odd prime number always.

Number the ordinals of odd points at seriate each [RLS.sub.No1~NoX+y] by consecutive natural numbers which begin with 1, namely from left to right each odd point at seriate each [RLS.sub.No1~NoX+y] is marked with from small to great a natural number [greater than or equal to]1 in the proper order, where y [greater than or equal to]0.

Then, there is one No(X+y) kind's odd point within [J.sub.x+y] odd points which share an ordinal at [J.sub.x+y] [RLSS.sub.No1~No(x+y-1)] of a [RLS.sub.No1~Nox+y].

Furthermore, there is one No(X+y) kind's odd composite point within [J.sub.x+y] odd points which share an ordinal at [J.sub.x+y] [RLSS.sub.No1~No(x+y-1)] of seriate each [RLS.sub.No1~Nox+y] on the right of No1 [RLS.sub.No1~Nox+y].

Odd prime points [J.sub.1], [J.sub.2] ... [J.sub.x-1] and [J.sub.x] are at No1 [RLS.sub.No1~NoX]. Yet, there are X odd composite points on ordinals of [J.sub.1] plus [J.sub.2] ... plus [J.sub.x-1] plus [J.sub.x] at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX]. Thus No1 [RLS.sub.No1~NoX] is a particular [RLS.sub.No1~NoX] in contradistinction to each of others.

After change [??]s for * s at places of [SIGMA]NoX [X[greater than or equal to] 1] kind's odd composite points at the half line, if one * is separated from another * by [mu] *s plus b [??]s irrespective of their permutation, then express such a combinative form as a set of * [mu](*s)+b([??]s) *, where [mu] [greater than or equal to] 0, and b [greater than or equal to] 0.

If [mu]+2 *s of * [mu](*s)+b([??]s) * are all defined as odd prime points, then the set of * [mu](*s)+b([??]s) * is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Further, if the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] lies within consecutive [J.sub.x] odd points, and for odd prime number [J.sub.x], a number of residue's classes which [mu]+2 odd prime numbers whereof [mu]+2 [??]s express divided respectively by modulus [J.sub.x] is less than [J.sub.x], then, such a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the very a set of n-odd prime points, where n=[mu]+2.

If two *s of * v([??]s) * are defined as odd prime points, then the pair of *v([??]s)* is rewritten as a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where v [greater than or equal to] 0.

When [mu]=0, a set of * [mu](*s)+b([??]s) * is exactly a pair of *b([??]s)*, and a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is exactly a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where b [greater than or equal to] 0.

Let [mu]+b=m, a set of * [mu](*s)+b([??]s) * may be written as a set of * m(*s[??]s) *, and a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] may be written as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

After change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, [J.sub.x-h] at No1 [RLS.sub.No1~NoX] is defined as an odd prime point, where X>h [greater than or equal to] 0, yet there are infinitely many *s on the right of [J.sub.x] at the half line, and every * is an undefined odd point on prime/composite attribute. Anyhow every prime factor of an odd number which each * at the right of [J.sub.x] expresses is greater than [J.sub.x].

A set of * [mu](*s)+b([??]s) * is negated according as any * of the set is defined as one Also a pair of * v([??]s) * is negated according as either * of the pair is defined as one If a set of * [mu](*s)+b([??]s) * can not always be negated, then it is precisely a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Likewise, if a pair of * v([??]s) * can not always be negated, then it is precisely a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the definition for recurring segments of X kinds' odd points, we can conclude that after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, if there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.x] odd points on the right of [J.sub.x] at No1 [RLS.sub.No1~NoX], then there is surely a set of * [mu](*s)+b([??]s) * on ordinals of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX].

Without doubt, the converse proposition is tenable too. Namely after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, if there is a set of * [mu](*s)+b([??]s)* within consecutive [J.sub.x] odd points at seriate each [RLS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX], and from left to right Nok odd prime points of all sets of *[mu](*s)+b([??]s) * share an ordinal, then there is surely a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ordinals of any such set of * [mu](*s)+b([??]s) *, at No1 [RLS.sub.No1~NoX], where k = 1, 2, ... [mu]+2.

Of course, every [??] of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and every prime factor of an odd number which each * of every such set of * [mu](*s)+b([??]s) * expresses are greater than [J.sub.x].

To be brief, after change [??]s for *s at places of [SIGMA]NoX [X [greater than or equal to] 1] kind's odd composite points, a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.x] at No1 [RLS.sub.No1~NoX] and infinite many sets of * [mu](*s)+b([??]s)* on ordinals of the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate [RLSS.sub.No1~NoX] on the right of No1 [RLS.sub.No1~NoX] coexist at the half line.

We term the aforesaid conclusion as the coexisting theorem of a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus infinite many sets of * [mu](*s)+b([??]s)* at the half line, or term it as the coexisting theorem for short.

[J.sub.x+1] [RLSS.sub.No1~NoX] of any [RLS.sub.No1~No(x+1)] may be folded at an illustration, one by one, so as to view conveniently, e.g. No1, No2 and No3 kinds' odd points at two [RLSS.sub.No1~No3] from the differentia, please, see following fourth illustration.

[ILLUSTRATION OMITTED]

After change [??]s for *s at places of No1 plus No2 plus No3 kinds' odd composite points, every [??] denotes a definite odd prime point, and every * denotes an undefined odd point at prime/composite attribute, and every [??] denotes a definite odd composite point, in the illustration. Line segment 3(211) is No1 [RLS.sub.No1~No3] and line segment CD is any of seriate [RLSS.sub.No1~No3] on the right of No1 [RLS.sub.No1~No3].

The Proof

We will prove together that there are infinitely many sets of n-odd prime numbers and pairs of consecutive odd prime numbers by the mathematical induction with the aid of [RLSS.sub.No1~NoX] and odd points thereof, thereinafter.

1. When X=1, there is a set of [??][??] alone on the right of [J.sub.1] at No1 [RLS.sub.No1], and the set of [??][??] is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as well, i.e. twin odd prime points 5 and 7, where [v.sub.1]=0.

When X=2, there are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.2] at No1 [RLS.sub.No1~No2], and these odd points contain several sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.s] odd points, including several pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within them, where [[mu].sub.2] [less than or equal to] 6, [b.sub.2] [less than or equal to] 5, [J.sub.1] [less than or equal to] [J.sub.s] [less than or equal to] [J.sub.5], and [v.sub.2] [less than or equal to] 2.

Evidently these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] contain pairs of twin odd prime points.

When X=3, there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.f] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.3] at No1 [RLS.sub.No1~No3], where [[mu].sub.2] [less than or equal to] [[mu].sub.3] [less than or equal to] 41, [b.sub.2] [less than or equal to] [b.sub.3] [less than or equal to] 58, [J.sub.s] [less than or equal to] [J.sub.f] [less than or equal to] [J.sub.27] =101, and [v.sub.2] [less than or equal to] [v.sub.3] =0, 1, 2, 3, 4, 5 and 6.

Evidently these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody certain sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody all pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on values of [v.sub.3] at No1 [RLS.sub.No1~No3], we instance (11, 13), (13, 17), (23, 29), (89, 97), (139, 149), (199, 211) and (113, 127). Please, see preceding third illustration once again.

When X=4, there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.a] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.4] at No1 [RLS.sub.No1~No4], where [[mu].sub.3] [less than or equal to] [[mu].sub.4] [less than or equal to] 337, [b.sub.3] [less than or equal to] [b.sub.4] < 813, [J.sub.f] [less than or equal to] [J.sub.a] [less than or equal to] [J.sub.189] =1151, and [v.sub.3] [less than or equal to] [v.sub.4] =0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 16.

Evidently these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody certain sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody all pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on values of [v.sub.4] at No1 [RLS.sub.No1~No4], we instance (17, 19), (19, 23), (31, 37), (89, 97), (139, 149), (211, 223), (293, 307), (1831, 1847), (1259, 1277), (887, 907), (1669, 1693), (2179, 2203) and (1327, 1461). Please, see preceding third illustration once more.

2. When X=[beta][greater than or equal to]4, suppose that there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.b] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[beta]] at No1 [RLS.sub.No1~No[beta]], where [[mu].sub.[beta]] [greater than or equal to] [[mu].sub.4,], [b.sub.[beta]] [greater than or equal to][b.sub.4], v[beta] [greater than or equal to] v4, [J.sub.b] [greater than or equal to] [J.sub.a], and [J.sub.[beta]] [greater than or equal to] [J.sub.4]. In addition, these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody any of sets of n-odd prime points on the right of [J.sub.1] at No1 [RLS.sub.No1~No[psi]], and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] embody any of pairs of consecutive odd prime points at No1 [RLS.sub.No1~No[psi]], where [psi]<[beta].

Let us suppose that any of sets of n-odd prime points on the right of [J.sub.1] at No1 [RLS.sub.No1~No[psi]] is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ; and any of pairs of consecutive odd prime points at No1 [RLS.sub.No1~No[psi]] is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where [[mu].sub.p] [greater than or equal to] [[mu].sub.4], [b.sub.q] [greater than or equal to] [b.sub.4], and [v.sub.[delta]] [greater than or equal to] [v.sub.4].

3. When X=[eta]>[beta], prove that there are both sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.c] odd points and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[eta]] at No1 [RLS.sub.No1~No[eta]], where [[mu].sub.[eta]] [greater than or equal to] [[mu].sub.[beta]], [b.sub.[eta]] [greater than or equal to] b[beta], v[eta] [greater than or equal to] [v.sub.[beta]] [J.sub.c] [greater than or equal to] [J.sub.b], and [J.sub.[eta]] > [J.sub.[beta]]. In addition, these sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must embody a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove, and these pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must embody a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove.

Proof . Since there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within consecutive [J.sub.b] odd points on the right of [J.sub.[beta]] at No1 [RLS.sub.No1~No[beta]], furthermore, we name the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]", where d [greater than or equal to] 1 and g =[beta]+d+[[mu].sub.p]+1. Well then, let us first prove that there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ordinals of [J.sub.[beta]]+d [[mu].sub.p]([??]s)+bq([??]s) [J.sub.g] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog], hereinafter.

We know that every odd number >1 has a smallest prime factor except for 1 surely, yet the smallest prime factor of any odd prime number is exactly it itself.

If greatest one within respective smallest prime factors of [b.sub.q] odd composite numbers whereof [b.sub.q]([??]s) between [J.sub.[beta]+d] and [J.sub.g] express is written as [J.sub.[phi]], then the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is either at No1 [RLS.sub.No1~No[phi]] or out of No1 [RLS.sub.No1~No[phi]]. If it is at No1 [RLS.sub.No1~No[phi]], then let 1 [less than or equal to] X1 [less than or equal to] [phi]. If it is out of No1 [RLS.sub.No1~No[phi]], then suppose that it is just at No1 [RLS.sub.No1~No[kappa]], but it is not at No1 [RLS.sub.No1~No[kappa]-1] then [kappa] > [phi], and let 1[less than or equal to] X2 [less than or equal to][kappa].

If the set of [J.sub.[beta]+d] [[mu].sub.p] ([??]s)+[b.sub.q]([??]s) [J.sub.g] is at No1 [RLS.sub.No1~No[phi]] then after change [??]s for *s at places of [SIGMA][No.sub.X1][1 [less than or equal to] X1 [less than or equal to] [phi]] kind's odd composite points, there is a set of *[[mu].sub.p](*s)+[b.sub.q]([??]s)* on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] at seriate each [RLS.sub.No1~No[phi]] on the right of No1 [RLS.sub.No1~No[phi]].

If the set of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] is just barely at No1 [RLS.sub.No1~No[kappa]], but it is out of No1 [RLS.sub.No1~No([kappa-1]], then after change [??]s for *s at places of [SIGMA][No.sub.X2] [1 [less than or equal to] X2 [less than or equal to] [kappa]] kind's odd composite points, there is a set of * [[mu].sub.p](*s)+[b.sub.q]([??]s)* on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] at seriate each [RLS.sub.No1~No[kappa]] on the right of No1 [RLS.sub.No1~No[kappa]].

Either [J.sub.[phi]] or [J.sub.[kappa]] [greater than or equal to] [[mu].sub.p] + [b.sub.q] +2, uniformly let it to equal [J.sub.v]. If [J.sub.[phi]] or [J.sub.[kappa]] < [[mu].sub.p] + [b.sub.q] +2, then suppose that [J.sub.v] is the smallest odd prime number which is not smaller than [[mu].sub.p] + [b.sub.q] +2.

Each set of *[[mu].sub.p] (*s)+[b.sub.q]([??]s) * on ordinals of [J.sub.[beta]+d] [[mu].sub.p]([??]s)+[b.sub.q]([??]s)[J.sub.g] considering aforementioned either case is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If some set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , then the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is rewritten as a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let v+1 [less than or equal to][omega][less than kor equal to]g, since there is one No[omega] kind's odd point within consecutive [J.sub.[omega]] odd points, and there is one No[omega] kind's odd point within [J.sub.[omega]] odd points which share an ordinal at seriate [J.sub.[omega]] [RLSS.sub.No1~No[omega]-1], therefore there is a series of results as the following.

After successively change [??]s for *s at places of No (v+1) kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p]) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p] sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No~No(v+1)] on the right of No1 [RLS.sub.No1~No(v+1)]. Of course, every prime factor of an odd number which each [??] at here expresses is greater than [J.sub.v+1].

After successively change [??]s for *s at places of No (v+2) kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p]) ([J.sub.v+2] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p]([J.sub.v+2] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~No(v+1)] on the right of No1 [RLS.sub.No1~No(v+1)]. Of course, every prime factor of an odd number which each [??] at here expresses is greater than [J.sub.v+2].

And so on and so forth ...

Up to after successively change [??]s for *s at places of Nog kind's odd composite points, there are both ([J.sub.v+1] - [[mu].sub.p])([J.sub.v+2] - 1)([J.sub.v+3] - 1)...([J.sub.g] - 1) sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[mu].sub.p]([J.sub.v+2] - 1)([J.sub.v+3] - 1)...([J.sub.g] - 1) pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at seriate each [RLS.sub.No1~Nog] on the right of No1 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Of course, every prime factor of an odd number which each [??] at here expresses is greater than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the half line on the right of No1 [RLS.sub.No1~Nog] has infinitely many [RLSS.sub.No1~Nog], thus there are both infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and infinitely many pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the half line after successively change [??]s for *s at places of [SIGMA]No[omega] [v+1 [less than or equal to] [omega] [less than or equal to] g] kind's odd composite points. Concurrently, there are infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ) infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ... and infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the half line. Of course, every prime factor of an odd number which each [??] within aforementioned sundry sets expresses is greater than [J.sub.g].

Thus there are a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , ... a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , and a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog] according to aforesaid that coexisting theorem.

Thus it can seen, preceding results contain such a conclusion, namely there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog]. This is just the proposition which need us to prove.

In case a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove is embodied within the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... plus the set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus the pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog], the pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is proven synchronously into the real too.

If a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which needs us to prove is not embodied within aforementioned sundry sets of n-odd prime points, then we can likewise apply the aforesaid way of doing according to the coexisting theorem to prove and get that there is a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which embodies such a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.g] at No1 [RLS.sub.No1~Nog,] but values of g on two places are perhaps unlike.

Since the mathematical induction sets up a claim to X=[eta]>[beta], whereas now has X=g=[beta]+d+[[mu].sub.p]+1> [beta], thus can replace g by [eta], therefore, we have proven that there is a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the right of [J.sub.[eta]] at No1 [RLS.sub.No1~No[eta]].

When vest further x with a value which is greater than g, we likewise can continue to apply the aforesaid way of doing and the coexisting theorem to prove and get that thre are another set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and another pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . And so on and so forth ...

Though values of X are not consecutive natural numbers under the prerequisite that it is proven there are sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the aforesaid way of doing and the coexisting theorem, but, since there are infinitely many natural numbers at all events, so that there are infinitely many values of X which accord with the claim. Therefore there are both infinitely many sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and infinitely many pairs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since a set of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expresses a set of n-odd prime numbers, where n=[[mu].sub.p]+2, consequently there are infinitely many sets of n-odd prime numbers.

In addition, a pair of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expresses a pair of consecutive odd prime numbers which differ by 2([v.sub.[delta]]+1), consequently there are infinitely many pairs of consecutive odd prime numbers which differ by 2k, where k=[v.sub.[delta]]+1.

Taken one with another, we have proven that there are both infinitely many sets of n-odd prime numbers and infinitely many pairs of consecutive odd prime numbers which differ by 2k.

Zhang Tianshu

Nanhai west oil corporation, China offshore oil, Zhanjiang city, Guangdong province, P.R. China.

Email: (1) tianshu_zhang507@yahoo. com. cn;

(2) friend_zhang888@sina.com; (3) xinshijizhang@hotmail.com

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Author: | Tianshu, Zhang |
---|---|

Publication: | Advances in Theoretical and Applied Mathematics |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | May 1, 2013 |

Words: | 5585 |

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