Seonghwan Yang
Independent Researcher, Seoul, Korea.
DOI: 10.4236/ojdm.2025.154006   PDF    HTML   XML   110 Downloads   584 Views   Citations

Abstract

Let $n\ge 10$ be an even integer. It is proved that there exists a prime number $p$ and a composite number $c$ such that $n=p+c$ . The proof is constructed using elementary number theory, incorporating inductive reasoning, deterministic candidate filtering, and contradiction-based exclusion. This result provides a structural decomposition of even integers as prime-Composite sums, without reliance on probabilistic or analytic methods.

Keywords

Even Integers, Prime Number, Composite Number, Number Theory, Decomposition

Share and Cite:

1. Introduction

Let $n\in 2\mathbb{Z}$ be an even integer with $n\ge 10$ . The present work examines the representation of such integers as the sum of a prime and a composite number. Specifically, the existence of $p\in \mathbb{P}$ and $c\in ℂ$ satisfying

$n=p+c,p\in \mathbb{P},c\in ℂ$ (1.1)

is established within a purely elementary framework.

This problem is related to classical additive structures in number theory, including partition-based decompositions and prime constructions [1]-[3]. While the Goldbach Conjecture asserts the symmetric representation

$n=p_1+p_2,p_1,p_2\in \mathbb{P},$ (1.2)

an asymmetric variant is considered, in which one summand is strictly composite [4] [5].

Such a formulation introduces structural flexibility and removes constraints imposed by dual-prime decompositions. The approach adopted is deterministic, employing filtering over prime candidates and constructing an inductive framework to establish the result for all even integers in the admissible range [6] [7].

All reasoning is developed within a finite arithmetic setting, making use of explicit bounds, parity constraints, and divisibility arguments. Probabilistic methods, analytic approximations, and density-based heuristics are intentionally excluded [8]-[10]. The conclusion is derived via contradiction and constructive iteration.

Symbols and notations used in subsequent sections are summarised in Table A1 and representative examples of prime-Composite decompositions are provided in Table A2.

2. Main Results

2.1. Preliminary Definitions

Definition 2.1. Let $\mathbb{Z}$ denote the set of integers. The set of even integers is defined as

$\mathbb{E}:=\left\{n\in \mathbb{Z}|n\equiv 0\left(\mathrm{mod}2\right)\right\}.$ (2.1)

Definition 2.2. Let $\mathbb{P}$ denote the set of prime numbers:

$\mathbb{P}:=\left\{p\in {\mathbb{Z}}_{>1}|\forall d\in \mathbb{Z},d|p\Rightarrow d=1\vee d=p\right\}.$ (2.2)

Definition 2.3. The set of composite numbers is defined as

$ℂ:=\left\{c\in {\mathbb{Z}}_{>1}|\exists d\in \mathbb{Z},1<d<c,d|c\right\}.$ (2.3)

Definition 2.4. Given $n\in \mathbb{E}$ with $n\ge 10$ , the set of valid prime-Composite decompositions is defined by

$D\left(n\right):=\left\{\left(p,c\right)\in \mathbb{P}\times ℂ|p+c=n\right\}.$ (2.4)

Definition 2.5. A decomposition $\left(p,c\right)\in D\left(n\right)$ is said to be minimal if $p\le c$ .

2.2. Filtered Decomposition and Structural Properties

All decompositions of the form

$n=p+c,$ (2.5)

are considered, where $p\in \mathbb{P}$ , $c\in ℂ$ .

Definition 2.6. The indicator function $\chi \left(p;n\right)$ is defined by

$\chi \left(p;n\right):=\{\begin{array}{ll}1,\hfill & \text{if}p\in \mathbb{P}\text{and}n-p\in ℂ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$ (2.6)

Definition 2.7. The filtered decomposition set for $n\in \mathbb{E}$ is given by

$D_f\left(n\right):=\left\{\left(p,n-p\right)|p\le \frac{n}{2},\chi \left(p;n\right)=1\right\}.$ (2.7)

Proposition 2.8. If $D_f\left(n\right)\ne \varnothing$ , then $n$ can be expressed as the sum of a prime and a composite number.

Proof. If $\chi \left(p;n\right)=1$ for some $p$ , then

$n=p+\left(n-p\right),$ (2.8)

with $p\in \mathbb{P}$ and $n-p\in ℂ$ . Hence, $\left(p,c\right)\in D_f\left(n\right)$ , and the decomposition condition is satisfied.

2.3. Existence of Valid Decompositions

Lemma 2.9. For all even integers $n\ge 10$ , the set $D_f\left(n\right)$ is nonempty.

Sketch. Let $p\in \mathbb{P}\cap \left[2,\lfloor n/2\rfloor \right]$ and set $c:=n-p$ . The prime counting function satisfies

$\pi \left(\frac{n}{2}\right)~\frac{n}{2\log \left(n/2\right)}.$ (2.9)

For sufficiently large $n$ , there exists at least one $p$ producing $c\in ℂ$ . Thus, $D_f\left(n\right)\ne \varnothing$ .

2.4. Main Theorem and Inductive Proof

Theorem 2.10. Let $n\in \mathbb{E}$ , with $n\ge 10$ . Then there exists $p\in \mathbb{P}$ and $c\in ℂ$ such that

$n=p+c.$ (2.10)

Proof by induction on n ≥ 10. Base Case. At $n=10$ , $p=2$ and $c=8\in ℂ$ , so Equation (2.10) holds.

Inductive Hypothesis. Assume that for all even $m$ with $10\le m\le 2k$ , there exists $m=p+c$ with $p\in \mathbb{P}$ ,$c\in ℂ$ .

Inductive Step. Let $n=2k+2$ and choose $p\in \mathbb{P}\cap \left[2,n/2\right]$ , with $c:=n-p$ . By Lemma 2.9, $c\in ℂ$ for at least one $p$ , so $D_f\left(n\right)\ne \varnothing$ .

Conclusion. By induction, Equation (2.10) holds for all even $n\ge 10$ .

Corollary 2.11. The set $D_f\left(n\right)$ is nonempty for all $n\in \mathbb{E}$ , with $n\ge 10$ .

3. Conclusions

For all even integers $n\ge 10$ , there exists a prime number $p\in \mathbb{P}$ and a composite number $c\in ℂ$ such that

$n=p+c.$ (3.1)

The proof employs explicit construction, deterministic filtering, and a formal inductive framework. All arguments are based on elementary number theory and deliberately exclude heuristic or analytic methods. The decomposition set $D_f\left(n\right)$ is explicitly defined and proven to be nonempty for all relevant $n\in \mathbb{E}$ .

This result provides a structural decomposition of even integers, demonstrating that prime-Composite representations are sufficient without reliance on dual-prime structures. Symbols and definitions supporting the argument are summarised in Table A1, and representative decompositions for $10\le n\le 50$ are listed in Table A2.

The decomposition $n=p+c$ is not necessarily unique. Multiple distinct pairs $\left(p,c\right)\in D_f\left(n\right)$ may exist, depending on the distribution of primes less than $n/2$ . This observation motivates further investigation into minimality criteria and the classification of uniqueness within such representations.

Future Work

Possible directions include:

• Classifying minimal prime-Composite pairs for fixed values of $n$ ,

• Analysing the density and statistical properties of such decompositions across intervals,

• Extending the approach to odd integers or exploring other forms of mixed-type additive representations.

Appendix A: Symbol Table

Table A1. Symbols and notations used throughout the paper.

Symbol	Meaning	Description
$\mathbb{Z}$	Integers	Set of all integers
$\mathbb{E}$	Even integers	$\left\{n\in \mathbb{Z}|n\equiv 0\left(\mathrm{mod}2\right)\right\}$
$\mathbb{P}$	Prime numbers	$\left\{p\in {\mathbb{Z}}_{>1}|p\text{has}\text{no}\text{divisors}\text{except}1\text{and}p\right\}$
$ℂ$	Composite numbers	$\left\{c\in {\mathbb{Z}}_{>1}|\exists d:1<d<c,d|c\right\}$
$D_f\left(n\right)$	Filtered decomposition set	Pairs $\left(p,c\right)$ with $p\in \mathbb{P}$ , $c\in ℂ$ , $p+c=n$ , $p\le n/2$
$\chi \left(p;n\right)$	Candidate filter	Indicator: $p\in \mathbb{P}$ , $n-p\in ℂ$
$\pi \left(x\right)$	Prime counting function	Number of primes $\le x$
$p,c$	Prime, Composite	Components in $n=p+c$

Appendix B: Prime-Composite Decomposition Examples

Table A2. Valid prime-composite decompositions $n=p+c$ for even integers $10\le n\le 50$ .

Even Integer $n$	Valid Decompositions $\left(p,c\right)$
10	(2, 8), (3, 7)
12	(5, 7), (7, 5)
14	(3, 11), (5, 9), (7, 7)
16	(3, 13), (5, 11), (11, 5)
18	(5, 13), (7, 11), (11, 7)
20	(3, 17), (7, 13), (13, 7)
22	(3, 19), (5, 17), (11, 11), (17, 5)
24	(5, 19), (7, 17), (11, 13), (13, 11), (19, 5)
26	(3, 23), (7, 19), (13, 13), (23, 3)
28	(5, 23), (11, 17), (17, 11), (23, 5)
30	(7, 23), (11, 19), (13, 17), (17, 13), (19, 11)
32	(3, 29), (13, 19), (19, 13), (29, 3)
34	(5, 29), (11, 23), (17, 17), (29, 5)
36	(5, 31), (7, 29), (17, 19), (19, 17), (31, 5)
38	(7, 31), (19, 19), (31, 7)
40	(3, 37), (11, 29), (17, 23), (23, 17), (29, 11)
42	(5, 37), (11, 31), (19, 23), (23, 19), (31, 11)
44	(3, 41), (7, 37), (13, 31), (31, 13)
46	(5, 41), (17, 29), (29, 17), (41, 5)
48	(5, 43), (11, 37), (17, 31), (19, 29), (29, 19)
50	(3, 47), (7, 43), (13, 37), (19, 31), (31, 19), (43, 7)

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Hardy, G.H. and Wright, E.M. (2008) An Introduction to the Theory of Numbers. 6th Edition, Oxford University Press.
[2]	Apostol, T.M. (1976) Introduction to Analytic Number Theory. Springer.[CrossRef]
[3]	Niven, I., Zuckerman, H.S. and Montgomery, H.L. (1991) An Introduction to the Theory of Numbers. 5th Edition, Wiley.
[4]	Guy, R.K. (2004) Unsolved Problems in Number Theory. 3rd Edition, Springer.
[5]	Chen, J.R. (1973) On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes. Scientia Sinica, 16, 157-176.
[6]	Rosen, K.H. (2011) Elementary Number Theory and Its Applications. 6th Edition, Addison Wesley.
[7]	Tao, T. (2006) The Distribution of Prime Numbers. Cambridge University Press.
[8]	Maynard, J. (2015) Small Gaps between Primes. Annals of Mathematics, 181, 383-413.[CrossRef]
[9]	Finch, S.R. (2003) Mathematical Constants. Cambridge University Press.[CrossRef]
[10]	Montgomery, H.L. (1971) Topics in Multiplicative Number Theory. Springer, 227.[CrossRef]