On the Representation of a Number as the Sum of Two Squares and a Prime

Shihui You

doi:doi:10.11648/j.pamj.20261502.12

Research Article |

| Peer-Reviewed

On the Representation of a Number as the Sum of Two Squares and a Prime

Shihui You^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 2)

Received: 23 March 2026 Accepted: 1 April 2026 Published: 20 April 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 2)
DOI	10.11648/j.pamj.20261502.12
Page(s)	18-28
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Diophantine Equation, Number of Solutions, Set of Solutions, Subsets of Integer, Combinatorial Approximation Method, Enumeration Method

1. Introduction and Results

1.1. Introduction

Hooley, based on the Generalized Riemann Hypothesis (GRH)

[1]

, provided proof of a conjecture proposed by Hardy and Littlewood

[2]

. Considering an integer

N

represented in the following form.

N = p + x_{1}^{2} + x_{2}^{2}

(1)

where

p

is a prime,

x_{1}

x_{2}

is a nonzero integer, and when

N \to \infty, N

the asymptotic formula for the number of such representations is

π S (N) Li (N)

(2)

where

S (N) = \prod_{p} (1 + \frac{χ - 4 (p)}{p (p - 1)}) \prod_{p | N} \frac{(p - 1) (p - χ - 4 (p))}{p^{2} - p + χ - 4 (p)}

or, when odd

N \to \infty, N

the asymptotic formula for the number of such representations is given by Ricardo Adonis Caraccioli Abrego

[3].

S (N) c_{\infty} \frac{N}{lnN}

(3)

where

S (N) > 0, c_{\infty} > 0

In particular, every sufficiently large integer can be represented in this form. In 1960, Linnik

[4]

obtained the first unconditional proof of the Hardy-Littlewood problem using his dispersion method, thereby removing the assumption of the Generalized Riemann Hypothesis. Following the introduction of the Bombieri-Vinogradov theorem, the proof was greatly simplified. See

[5]

The problem of the representation of an integer as this form

N = p + x_{1}^{2} + x_{2}^{2}

can be transferred to finding the number of integer solutions of the equation

p + x_{1}^{2} + x_{2}^{2} = N

. Obtaining solutions to general Diophantine equations is very difficult, whereas determining the number of solutions to such equations is comparatively much easier. Combinatorial methods, combinatorial approximation methods, enumeration methods, and the density transformation method can be employed to determine the number of integer solutions to the Diophantine equations with countable solution sets. By these methods, the following results can be established.

1.2. Results

Theorem 1 Equation

p + x_{1}^{2} + x_{2}^{2} = N

(4)

where

p

is a prime,

x_{1}

x_{2}

is nonzero integer. When the integer

N \to \infty

, the approximate number of integer solutions (NS) of the equation

p + x_{1}^{2} + x_{2}^{2} = N

is expressed as

NS ~ \frac{cN}{lnN}

(5)

where

c

is a constant,

c = \frac{\sum_{i = 1}^{k} (\frac{1}{lnk})}{2 \sum_{i = 1}^{2 k} (\frac{1}{lnk})} > \frac{1}{4}

, which can be computed to any desired accuracy.

To prove theorem 1, it is necessary to study Lemmas 1-3. In order to obtain the number of integer solutions of equation

p + x_{1}^{2} + x_{2}^{2} = N

, the problem of determining the number of integer solutions to general Diophantine equations is transformed into finding the number of integer solutions to linear Diophantine equations whose countable solution sets are integer subsets or sequences of integer subsets.

Equation

p + x_{1}^{2} + x_{2}^{2} = N

, its solution set is

p = \{2, 3, 5, \dots, p_{i}, \dots, N\}

, and solution set

x_{i} (i = 1, 2) = \{1, 2, 3, \dots, j, \dots, N\}

Let substitute

p = y_{1}

x_{1}^{2} = y_{2}

and

x_{2}^{2} = y_{3}

, transforming it into the following equation.

y_{1} + y_{2} + y_{3} = N

(6)

Its solution set is

y_{1} = p = \{2, 3, 5, \dots, N\}

, and

{y_{2} = y}_{3} = x_{1}^{2} = x_{2}^{2} = \{1^{2}, 2^{2}, 3^{2}, \dots, j^{2}, \dots, N\}

For determining the number of integer solutions of the equation such as

p + x_{1}^{2} + x_{2}^{2} = N

, first, we introduce the combinatorial method, approximate combinatorial method, and enumeration method to compute the number of integer solutions of linear Diophantine equations whose the solution set is integer set (integer sequence), and is countable.

Second, for cases where the solution sets and the number of solution sets of the variables in linear Diophantine equations are either the same or different, we propose the approximate combinatorial method, enumeration method, and density transformation method to compute an approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets.

2. Proof of Lemma

2.1. Lemma

Why study linear Diophantine equations? More specifically, why analyze the number of integer solutions of linear Diophantine equations whose solution sets consist of sequences of integer subsets?

1) The number of integer solutions to linear Diophantine equations can be determined by combinatorial methods.

2) To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for linear Diophantine equations is comparatively easier.

3) The problem of counting the number of integer solutions to nonlinear or higher-order Diophantine equations can be transformed into the problem of counting the number of integer solutions to linear Diophantine equations whose solution sets are integer subsets.

4) The enumeration method allows for straightforward listing of all integer solutions of linear Diophantine equations when the solution set is an integer sequence, or all of integer solutions are expressed as numerical equations or equalities.

5) Since all integer solutions of linear Diophantine equations with solution sets formed by sequences of integer subsets belong to the integer solutions of linear Diophantine equations with integer sequences as solution sets, the enumeration method likewise facilitates the straightforward listing of all such solutions as numerical equations or equalities.

6) By utilizing the enumeration method, it is straightforward to list all local integer solutions, or all of equations or equalities of local integer solutions of linear Diophantine equations whose solution sets are sequences of integer subsets. Specifically, linear Diophantine equations can explicitly or concisely be represented by the numerical equivalence relations within each local equation (

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N))

This section mainly discusses the number of integer solutions to linear Diophantine equations whose countable solution sets are given by integer sets or integer subset (sequences). The definitions and notations used herein are consistent with those in reference

[6]

Lemma 1 presents the number of integer solutions of linear Diophantine equations with integer set sequence as the solution set. Lemma 2 provides the approximate number of integer solutions of linear Diophantine equations with integer set sequence as the solution set. Lemma 3 provides the approximate number of integer solutions of linear Diophantine equations when the solution sets and the number of the solution sets of variables are the same or different. Lemma 4 and Lemma 5 present the properties of general linear Diophantine equations when transformed into standard linear Diophantine equations.

Lemma 1 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

The number of integer solutions is well known as

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) = ​ (\begin{matrix} N - 1 \\ m - 1 \end{matrix}) = C_{N - 1}^{m - 1} = \frac{1}{(m - 1)!} \frac{N (N - 1) (N - 2) \dots (N - (m - 1))}{N}

For any non-negative integer, the solution set is

x_{i} = \{1, 2, 3, \dots, N\}

, and the number of the solution set is

|x_{i}| = N

Lemma 2 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

then the approximate number of integer solutions to the equation is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) \sim \frac{1}{(m - 1)!} \frac{1}{N} N^{m}

For any non-negative integer

N

, the solution set is

x_{i} = \{1, 2, 3, \dots, N\}

, and the number of the solution set is

|x_{i}| = N

Lemma 3 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

then the approximate number of integer solutions to the equation is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) \sim \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

For any non-negative integer

N

, the solution set is

x_{i} \subseteq \{1, 2, 3, \dots, N\}

, the solution set can be

x_{i} \neq x_{j}

, and the number of the solution set is

|x_{i}| \leq N

, it can be

|x_{i}| \neq |x_{j}|

Lemma 4 The number of integer solutions of equation

\pm x_{1} \pm x_{2} \pm \dots \pm x_{i} \pm \dots \pm x_{m} = N

is equal to the number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

Lemma 5 The number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = 0

is equal to the number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

2.2. Proof of the Lemmas

Lemma 1 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

The number of integer solutions is well known as

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) = (\begin{matrix} N - 1 \\ m - 1 \end{matrix}) = C_{N - 1}^{m - 1} = \frac{1}{(m - 1)!} \frac{N (N - 1) (N - 2) \dots (N - (m - 1))}{N}

For any non-negative integer

N

, the solution set

x_{i} = \{1, 2, 3, \dots, N\}

, the number of the solution set

|x_{i}| = N

Proof

We consider the number of integer solutions of the linear Diophantine equation where the solution set is an integer set or a sequence of integer set, and a countable solution set. The combinatorial method and the enumeration method are applied.

1. Combinatorial Method

Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

(7)

It is well known that the number of integer solutions is

[7]

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) = (\binom{N + m - 1}{m - 1}) = C_{N + m - 1}^{m - 1}

(8)

For any non-negative integer

N

, the solution set is

x_{i} = \{1, 2, 3, \dots, N\}

, and the number of the solution set is

|x_{i}| = N

Substitute

x_{i} - 1 = y_{i}

into equation (7) to obtain

y_{1} + y_{2} + \dots + y_{i} + \dots + y_{m} = N - m

(9)

The number of integer solutions of equation (9) is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) = (\binom{N - 1}{m - 1}) = C_{N - 1}^{m - 1}

(10)

The number of integer solutions of equation (9) is equal to the number of integer solutions of equation (7).

C_{m + N - 1}^{m - 1}

C_{N - 1}^{m - 1}

(\begin{matrix} N + m - 1 \\ m - 1 \end{matrix})

(\begin{matrix} N - 1 \\ m - 1 \end{matrix})

(11)

According to combinatorial relations, we have

C_{N}^{m} = \frac{N}{m} C_{N - 1}^{m - 1}

C_{N - 1}^{m - 1} = \frac{m}{N} C_{N}^{m}

(12)

C_{N}^{m} - C_{N - 1}^{m} = C_{N - 1}^{m - 1}

(13)

Equation (13) presents the recursive relation between the

m

-dimensional linear Diophantine equation (

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

) and the

(m - 1)

-dimensional linear Diophantine equation (

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m - 1} = N

). This equation (13) also provides the recursive relation between the height

N

linear Diophantine equation (

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

) and the height

(N - 1)

linear Diophantine equation

{(x}_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N - 1)

. Equation (13) can be used, based on height

N

and dimension

m

, to prove the number of solutions of the linear Diophantine equation by mathematical induction.

2. Local-Global Principle

The number of integer solutions of equation (7) is denoted as

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) = (\begin{matrix} N - 1 \\ m - 1 \end{matrix}) = C_{N - 1}^{m - 1} = \frac{1}{(m - 1)!} \frac{N (N - 1) (N - 2) \dots (N - (m - 1))}{N}

(14)

The product

\frac{1}{m!} N (N - 1) (N - 2) \dots (N - (m - 1))

in the formula

C_{N}^{m} - C_{N - 1}^{m} = C_{N - 1}^{m - 1}

, represents the total number of integer solutions of the equation in

m

dimensions.

In the product

\frac{1}{m!} \frac{N (N - 1) (N - 2) \dots (N - (m - 1))}{N}

, the product

\frac{1}{m!} N (N - 1) (N - 2) \dots (N - (m - 1))

is divided by

N

, therefore for each

N (N = 1, 2, \dots, i, \dots, N)

, if the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

) has solutions, then the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

also has solutions. If the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

has no solutions, then, the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

has only finitely many solutions in some

i

3. Enumeration Method

Linear Diophantine equations can provide clear or concise numerical equivalence relations in each local equation (

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N))

In linear Diophantine equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N,

when

m = 3,

the equation is transformed into

x_{1} + x_{2} + x_{3} = N

According to lemma 5, the number of integer solutions to the equation

x_{1} + x_{2} + x_{3} = N

is equal to the number of integer solutions to the equation

x_{1} + x_{2} + x_{3} = 0

According to lemma 4, the number of integer solutions to the equation

x_{1} + x_{2} + x_{3} = 0

is equal to the number of integer solutions to the equation

x_{1} + x_{2} - x_{3} = 0

x_{1} + x_{2} = x_{3}

Now, consider the enumeration method

[8]

to list the integer solutions of linear Diophantine equations with a countable solution set.

From the above, when

m = 3,

the number of integer solutions to the equation

x_{1} + x_{2} = x_{3}

x_{1} + x_{2} + x_{3} = N

is equal to

C_{N - 1}^{3 - 1} = C_{N - 1}^{2}

Table 1 below shows the total number of integer solutions of the 3D equation

x_{1} + x_{2} = x_{3}

where

x_{i} \leq 10

Table 1. Total number of integer solutions of the 3D equation

x_{1} + x_{2} = x_{3}

where

x_{i} \leq 10

$x_{1} + x_{2} = x_{3}$	1	2	3	4	5	6	7	8	9
1	1+1=2	2+1=3	3+1=4	4+1=5	5+1=6	6+1=7	7+1=8	8+1=9	9+1=10
2	1+2=3	2+2=4	3+2=5	4+2=6	5+2=7	6+2=8	7+2=9	8+2=10
3	1+3=4	2+3=5	3+3=6	4+3=7	5+3=8	6+3=9	7+3=10
4	1+4=5	2+4=6	3+4=7	4+4=8	5+4=9	6+4=10
5	1+5=6	2+5=7	3+5=8	4+5=9	5+5=10
6	1+6=7	2+6=8	3+6=9	4+6=10
7	1+7=8	2+7=9	3+7=10
8	1+8=9	2+8=10
9	1+9=10
10
$x_{2}$

x_{1} = \{1, 2, 3, \dots, 10\}

x_{2} = \{1, 2, 3, \dots, 10\}

and

x_{3} = \{1, 2, 3, \dots, 10\}

The number of integer solutions to the equation

x_{1} + x_{2} = x_{3}

is equal to

C_{N - 1}^{3 - 1} = C_{N - 1}^{2} = \frac{(N - 1) (N - 2)}{2!}

(15)

When

N \leq 10

, as shown in Table 1.

When

m = 2

, the number of integer solutions to the equation

x_{1} + x_{2} = N

is equal to

C_{N}^{2} - C_{N - 1}^{2} = \frac{(N) (N - 1)}{2!} - \frac{(N - 1) (N - 2)}{2!} = C_{N - 1}^{1} = N - 1

(16)

is indicated by the diagonal in Table 1.

Lemma 1 is proved.

Lemma 2 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

be the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

the approximate number of integer solutions for the equation is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) \sim \frac{1}{(m - 1)!} \frac{1}{N} N^{m}

For any non-negative integer

N

, the solution set is

x_{i} = \{1, 2, 3, \dots, N\}

, and the number of the solution set is

|x_{i}| = N

Proof

1. Combinatorial approximation method

According to lemma 1, the equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

has the solution set

x_{i} = {1, 2, 3, \dots, N}

, and the number of the solution set is

|x_{i}| = N

The number of integer solutions for this equation is

C_{N}^{m} - C_{N - 1}^{m} = C_{N - 1}^{m - 1}

In the formula

C_{N}^{m} - C_{N - 1}^{m} = C_{N - 1}^{m - 1}

m

is finite and relatively

N

very small;

N

is very large and infinite.

The formula

C_{N}^{m} - C_{N - 1}^{m} = C_{N - 1}^{m - 1}

\frac{N (N - 1) \dots (N - (m - 1))}{m!} - \frac{(N - 1) (N - 2) \dots (N - m)}{m!} = \frac{(N - 1) (N - 2) \dots (N - (m - 1))}{(m - 1)!}

(17)

We make the following approximation

N (N - 1) \dots (N - (m - 1)) ~ N^{m}

(N - 1) (N - 2) \dots (N - m) ~ {〈N - 1〉}^{m}

and

(N - 1) (N - 2) \dots (N - (m - 1)) ~ N^{〈m - 1〉}

Obtain

\frac{N^{m}}{m!} - \frac{{(N - 1)}^{m}}{m!} ~ \frac{N^{(m - 1)}}{(m - 1)!}

(18)

2. Local-Global Principle

The number of solutions to equation (7) is denoted by

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) ~ \frac{1}{(m - 1)!} \frac{N^{m}}{N}

(19)

The product

\frac{N^{m}}{m!}

in the above formula (18) is the total number of solutions for the

m

-dimensional equation.

\frac{N^{m}}{m! N}

, the product

\frac{N^{m}}{m!}

is divided by

N

, therefore for each

N

(

N = 1, 2, \dots, i, \dots, N

), if the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

has solutions, then the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

also has solutions. If the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

has no solutions, then, the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

has only finitely many solutions in some

i

3. Enumeration Method

Equation

x_{1} + x_{2} = x_{3}

has the solution set

x_{1} = x_{2} = x_{3} = {1, 2, 3, \dots, N}

, and the number of solution set is

|x_{1}| = |x_{2}| = |x_{3}| = N

, and the density of the solution set is

d_{1} = d_{2} = d_{3} = 1

Table 2 lists the total number of solutions for the 3D equation

x_{1} + x_{2} = x_{3}

satisfies

x_{i} \leq 10

Table 2. Total Number of Solutions of the 3D Equation

x_{1} + x_{2} = x_{3}

where

x_{i} \leq 10

$x_{1} + x_{2} = x_{3}$	1	2	3	4	5	6	7	8	9
1	1+1=2	2+1=3	3+1=4	4+1=5	5+1=6	6+1=7	7+1=8	8+1=9	9+1=10
2	1+2=3	2+2=4	3+2=5	4+2=6	5+2=7	6+2=8	7+2=9	8+2=10	9+2=11
3	1+3=4	2+3=5	3+3=6	4+3=7	5+3=8	6+3=9	7+3=10	8+3=11	9+3=12
4	1+4=5	2+4=6	3+4=7	4+4=8	5+4=9	6+4=10	7+4=11	8+4=12	9+4=13
5	1+5=6	2+5=7	3+5=8	4+5=9	5+5=10	6+5=11	7+5=12	8+5=13	9+5=14
6	1+6=7	2+6=8	3+6=9	4+6=10	5+6=11	6+6=12	7+6=13	8+6=14	9+6=15
7	1+7=8	2+7=9	3+7=10	4+7=11	5+7=12	6+7=13	7+7=14	8+7=15	9+7=16
8	1+8=9	2+8=10	3+8=11	4+8=12	5+8=13	6+8=14	7+8=15	8+8=16	9+8=17
9	1+9=10	2+9=11	3+9=12	4+9=13	5+9=14	6+9=15	7+9=16	8+9=17	9+9=18
10
$x_{2}$

Solution set

x_{1}

x_{2}

and

x_{3}

x_{1} = \{1, 2, 3, \dots, 10\}

x_{2} = \{1, 2, 3, \dots, 10\}

and

x_{3} = \{1, 2, 3, \dots, 10\}

When

m = 3

, the number of integer solutions of the equation

x_{1} + x_{2} = x_{3}

x_{1} + x_{2} + x_{3} = N

is equal to

NS \sim \frac{N^{2}}{2!}

When

m = 2

, the number of integer solutions of the equation

x_{1} + x_{2} = N

is equal to

NS \sim \frac{N^{2}}{2!} - \frac{(N - 1)^{2}}{2!} \sim \frac{1}{1!} N^{1}

. shown by the diagonal line in Table 2.

The proof of Lemma 2 is completed.

Lemma 3 Let

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N)

denote the number of integer solutions of the linear Diophantine equation.

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

and the approximate number of integer solutions to the equation is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) \sim \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

For any non-negative integer

N

, the solution set is

x_{i} \subseteq \{1, 2, 3, \dots, N\}

, the solution set can be

x_{i} \neq x_{j}

, and the number of the solution sets is

|x_{i}| \leq N

, and it can be

|x_{i}| \neq |x_{j}|

Proof

1. Combinatorial approximation method

According to lemma 2, the equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

has the solution set

x_{i} = {1, 2, 3, \dots, N}

, and the number of the solution set is

|x_{i}| = N

, for the equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

, has the following approximate number of integer solutions:

NS \sim \frac{N^{m}}{m!} - \frac{(N - 1)^{m}}{m!} \sim \frac{1}{(m - 1)!} N^{(m - 1)}

(20)

We rewrite the above expression as

\frac{N^{m}}{m!} - \frac{(N - 1)^{m}}{m!} \sim \frac{1}{(m - 1)!} \frac{1}{N} N^{m}

(21)

The solution set is

x_{i} \subseteq {1, 2, 3, \dots, N}

and it can be

x_{i} \neq x_{j}

, and the number of the solution set is

|x_{i}| \leq N

, and it can be

|x_{i}| \neq |x_{j}|

, and the density of the solution set is

d_{i} = \frac{|x_{i}|}{N}

The above expression (21) multiplied by the density

(d_{1} ∙ d_{2} ∙ \dots ∙ d_{i} ∙ \dots ∙ d_{m})

of the solution set

(d_{1} ∙ d_{2} ∙ \dots ∙ d_{m}) \{\frac{N^{m}}{m!} - \frac{(N - 1)^{m}}{m!}\} \sim \frac{1}{(m - 1)!} \frac{1}{N} (d_{1} ∙ d_{2} ∙ \dots ∙ d_{m}) N^{m}

(22)

Leftside=

\frac{(d_{1} ∙ N) (d_{2} ∙ N) \dots (d_{m} ∙ N)}{m!} - \frac{(d_{1} ∙ N - d_{1}) (d_{2} ∙ N - d_{2}) \dots (d_{m} ∙ N - d_{m})}{m!}

(23)

Since

d_{i} \leq 1

, we make the following approximation

(d_{1} ∙ N - d_{1}) (d_{2} ∙ N - d_{2}) \dots (d_{m} ∙ N - d_{m}) ~ (d_{1} ∙ N - 1) (d_{2} ∙ N - 1) \dots (d_{m} ∙ N - 1)

The above formula can be approximated as

Left side

= \frac{(d_{1} ∙ N) (d_{2} ∙ N) \dots (d_{m} ∙ N)}{m!} - \frac{(d_{1} ∙ N - 1) (d_{2} ∙ N - 1) \dots (d_{m} ∙ N - 1)}{m!}

Left side

= \frac{|x_{1}| |x_{2}| \dots |x_{m}|}{m!} - \frac{(|x_{1}| - 1) (|x_{2}| - 1) \dots (|x_{m}| - 1)}{m!}

Right side

= \frac{1}{(m - 1)!} \frac{1}{N} (d_{1} ∙ d_{2} ∙ \dots ∙ d_{m}) N^{m} = \frac{1}{(m - 1)!} \frac{1}{N} (d_{1} ∙ N) (d_{2} ∙ N) \dots (d_{m} ∙ N)

Right side

= \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

Left side = Right side

\frac{|x_{1}| |x_{2}| \dots |x_{m}|}{m!} - \frac{(|x_{1}| - 1) (|x_{2}| - 1) \dots (|x_{m}| - 1)}{m!} ~ \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

(24)

Consider the non-uniform distribution of the number of integer solutions caused by the combination of density

d_{i}

in each

m

dimension with

N

change, and then

\frac{1}{(m - 1)!}

is modified to

c (m, d_{i})

c (m, d_{i})

can be obtained through calculation based on precision; that is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) ~ \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}| = c (m, d_{i}) \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

(25)

2. Local-Global Principle

The number of integer solutions of equation (7) is

NS = N (\underset{m}{\underset{⏞}{1, 1, \dots, 1}}; N) ~ \frac{1}{(m - 1) ！} \frac{1}{N} |x_{1}| |x_{2}| \dots |x_{m}|

(26)

The product

\frac{|x_{1}| |x_{2}| \dots |x_{m}|}{m!}

in the above formula (24) is the total number of integer solutions in dimension

m .

\frac{|x_{1}| |x_{2}| \dots |x_{m}|}{Nm!}

, the product

\frac{|x_{1}| |x_{2}| \dots |x_{m}|}{m!}

is divided by

N

N

(

N = 1, 2, \dots, i, \dots, N

), if the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

has solutions, then the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

also has solutions. If the global equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

has no solutions, then, the local equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = i (1 \leq i \leq N)

has only finitely many solutions in some

i

3. Enumeration Method

Example 1. Prime number

Solution set

p = \{2, 3, 5, \dots, p_{i}, \dots, N\}, p_{i} \leq N

, the number of prime solution sets is

|p| = \frac{N}{\ln N}

, and the density of the prime solution sets is

d_{1} = d_{2} = \frac{1}{\ln N}

. The approximate number of integer solutions of the equation

p_{1} + p_{2} = N

NS \sim c \frac{|p_{1}| \cdot |p_{2}|}{1! N} \sim \frac{0.62 N}{{(\ln N)}^{2}}

(including duplicate solutions). For details on Goldbach’s conjecture and the number of solutions of Goldbach’s equation

p_{1} + p_{2} = N

, please see reference

[6]

Table 3. Total number of integer solutions of 2D equation

p_{1} + p_{2} = N

where

N \leq 10

$p_{1} + p_{2} = N$	2	3	5	7

2	2+2=4
3		3+3=6	5+3=8	7+3=10

5		3+5=8	5+5=10	7+5=12

7		3+7=10	5+7=12	7+7=14



$p_{2}$

x_{i} = \{2, 3, 5, \dots, p_{i}, \dots, N\} \subset {1, 2, 3, \dots, N}

|x_{i}| \leq N = 10

Example 2. Residue class

In the equation

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{i} x_{i} + \dots + a_{m} x_{m} = N

, the solution set of the residue class is

a_{i} x_{i} = \{a_{i} \cdot 1, a_{i} \cdot 2, \dots, a_{i} \cdot k, \dots, N\}

By substitution

a_{i} x_{i} = y_{i}

we can obtain

y_{1} + y_{2} + \dots + y_{i} + \dots + y_{m} = N

Solution set

y_{i}

a_{i} x_{i} = \{a_{i} \cdot 1, a_{i} \cdot 2, \dots, a_{i} \cdot k, \dots, N\}

, therefore,

a_{i} \cdot k \leq N

, and the number of the solution set is

|a_{i} x_{i}| = \frac{N}{a_{i}}

In the equation

x_{1} + 2 x_{2} = x_{3}

, the solution sets

x_{1}

2 x_{2}

and

x_{3}

are

x_{1} = x_{3} = \{1, 2, 3, \dots, k, \dots, N\}

2 x_{2} = \{2, 4, \dots, 2 k, \dots, N\}

The number of the solution set

|x_{1}| = |x_{3}| = N, |2 x_{2}| = \frac{N}{2}

The number of integer solutions is

NS \sim \frac{|x_{1}| \cdot |2 x_{2}| \cdot |x_{3}|}{2! N} = \frac{N \cdot \frac{N}{2} \cdot N}{2 N} = \frac{N^{2}}{4}

Table 4. Total number of integer solutions of the equation

x_{1} + 2 x_{2} = x_{3}

, where

x_{1, 3} \leq 10

2 x_{2} \leq 10

$x_{1} + 2 x_{2} = x_{3}$	1	2	3	4	5	6	7	8	9

2	1+2=3	2+2=4	3+2=5	4+2=6	5+2=7	6+2=8	7+2=9	8+2=10	9+2=11

4	1+4=5	2+4=6	3+4=7	4+4=8	5+4=9	6+4=10	7+4=11	8+4=12	9+4=13

6	1+6=7	2+6=8	3+6=9	4+6=10	5+6=11	6+6=12	7+6=13	8+6=14	9+6=15

8	1+8=9	2+8=10	3+8=11	4+8=12	5+8=13	6+8=14	7+8=15	8+8=16	9+8=17

10
$2 x_{2}$

x_{i} \subseteq \{1, 2, 3, \dots, N\}, x_{i} \neq x_{j}, |x_{i}| \neq |x_{j}|, |x_{i}| \leq N = 10

4. Density Transformation Method

We now discuss the density transformation of linear Diophantine equations whose solution set is an integer subset.

Let

a_{1}

a_{2}

\dots

a_{m}

be positive integers to satisfy

\gcd (a_{1}, a_{2}, \dots, a_{m}) = 1

. In addition, let

NS = N (a_{1}, a_{2}, \dots, a_{m}; N)

be the number of integer solutions of the equation.

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{i} x_{i} + \dots + a_{m} x_{m} = N

(27)

For non-negative integers

x_{1}

x_{2}

\dots

x_{m}

. According to Schur's theorem, the approximate number of integer solutions of equation (27) is

NS = N (a_{1}, a_{2}, \dots, a_{m}; N) \sim \frac{N^{m - 1}}{(m - 1)! \cdot a_{1} \cdot a_{2} \cdot \dots \cdot a_{m}} (N \to \infty)

(28)

For non-negative integers

N

, provided that

N

is sufficiently large, especially when there exists an integer

N_{1}

such that each

N \geq N_{1}

has at least one representation

[7].

Equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

(29)

Non-negative integers

x_{1}

x_{2}

\dots

x_{m}

, and the solution sets are integer subsets.

x_{i} = \{{(a_{i})}_{1}, {(a_{i})}_{2}, {(a_{i})}_{3}, \dots, N\} \subseteq \{1, 2, 3, \dots, N\} (1 \leq i \leq m)

, and can be

x_{i} \neq x_{j}

. When

{(a_{i})}_{k} \leq N

, and the density of the solution set is

d_{i} = \frac{|x_{i}|}{N}

, and the number of the solution set is

|x_{i}| = d_{i} \cdot N \leq N

Transform

x_{i} = \frac{1}{d_{i}} \cdot y_{i}

, the solution set is

y_{i} = \{1, 2, 3, \dots, N\} (1 \leq i \leq m)

, and is the integer set.

Solution set

\frac{1}{d_{i}} \cdot y_{i} = \{\frac{1}{d_{i}} \cdot 1, \frac{1}{d_{i}} \cdot 2, \dots, \frac{1}{d_{i}} \cdot k, \dots, N\} \subseteq \{1, 2, 3, \dots, k, \dots, N\}

, and the number of the solution set is

|\frac{1}{d_{i}} \cdot y_{i}| = d_{i} \cdot N \leq N

, the solution set is

\frac{1}{d_{i}} \cdot y_{i}

, the density is

d_{i}

, and there are

\frac{1}{d_{1}} \cdot y_{1} + \frac{1}{d_{2}} \cdot y_{2} + \dots + \frac{1}{d_{i}} \cdot y_{i} + \dots + \frac{1}{d_{m}} \cdot y_{m} = N

(30)

According to formula (28), the approximate number of solutions of equation (30) is

NS = N (\frac{1}{d_{1}}, \frac{1}{d_{2}}, \dots, \frac{1}{d_{m}}; N) \sim \frac{m}{N} \frac{1}{(m)!} (d_{1} \cdot N) (d_{2} \cdot N) \cdot \dots \cdot (d_{m} \cdot N)

(31)

where

d_{i} \cdot N = |x_{i}|

. The approximate number of solutions of equation (30) is also

NS \sim \frac{|x_{1}| \cdot |x_{2}| \dots |x_{m}|}{N \cdot (m - 1)!} (N \to \infty)

(32)

Lemma 3 is proved.

For further discussion on the cases of unequal density

d_{i} \cdot N = |x_{i}|

and applications, see reference

[6]

Lemma 4 The number of integer solutions of the equation

\pm x_{1} \pm x_{2} \pm \dots \pm x_{i} \pm \dots \pm x_{m} = N

is equal to the number of integer solutions of the equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

Proof

In the equation

\pm x_{1} \pm x_{2} \pm \dots \pm x_{i} \pm \dots \pm x_{m} = N

(33)

By substituting

\pm x_{i} = y_{i}

we can obtain

y_{1} + y_{2} + \dots + y_{i} + \dots + y_{m} = N

(34)

The solution set of negative integers is

- x_{i} = \{- 1, - 2, - 3, \dots, - k, \dots, - N\}

, and when

- N \leq - k \leq - 1

, the number of the solution set is

|- x_{i}| = N

|- x_{i}| = |y_{i}|

According to lemma 2 and lemma 3, the approximate number of integer solutions of equation (33) can be expressed as

NS \sim \frac{|- x_{1}| \cdot |- x_{2}| \dots |- x_{m}|}{N \cdot (m - 1)!} ~ \frac{N^{m}}{N \cdot (m - 1)!} = \frac{N^{m - 1}}{(m - 1)!} (N \to \infty)

(35)

According to lemma 2 and lemma 3, the approximate number of integer solutions of equation (34) can be expressed as

NS \sim \frac{|y_{1}| \cdot |y_{2}| \dots |y_{m}|}{N \cdot (m - 1)!} ~ \frac{N^{m - 1}}{(m - 1)!} (N \to \infty)

(36)

The number of integer solutions of equation (33) is equal to the number of integer solutions of equation (34).

Proof of Lemma 4 is completed.

Lemma 5 The number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = 0

is equal to the number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

Proof

Equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = 0

(37)

becomes

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} + N = N

(38)

By substitution

x_{i} = y_{i}

(1 \leq i \leq m - 1)

, and

x_{m} + N = y_{m}

we can obtain

y_{1} + y_{2} + \dots + y_{i} + \dots + y_{m} = N

.(39)

Since

N

is a constant; therefore, it is not included the number of the solution set. The number of solution set

|x_{m} + N|

is only regarded as

|x_{m} + N|

|x_{m}|, or |x_{m} + N|

|x_{m}|

N .

According to lemma 2 and lemma 3, the approximate number of integer solutions of equation (38) can be expressed as

NS \sim \frac{|x_{1}| \cdot |x_{2}| \dots |x_{m} + N|}{N \cdot (m - 1)!} ~ \frac{N^{m}}{N \cdot (m - 1)!} = \frac{N^{m - 1}}{(m - 1)!} (N \to \infty)

(40)

According to lemma 2 and lemma 3, the approximate number of integer solutions of equation (39) can be expressed as

NS \sim \frac{|y_{1}| \cdot |y_{2}| \dots |y_{m}|}{N \cdot (m - 1)!} ~ \frac{N^{m - 1}}{(m - 1)!} (N \to \infty)

(41)

The number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = 0

is equal to the number of integer solutions of equation

x_{1} + x_{2} + \dots + x_{i} + \dots + x_{m} = N

Proof of Lemma 5 is completed.

3. Proof of Theorem 1

Theorem 1Equation

p + x_{1}^{2} + x_{2}^{2} = N

(42)

where

p

is prime,

x_{1}

x_{2}

are nonzero integer. When the integer

N \to \infty

, The approximate number of integer solutions (NS) of the equation

p + x_{1}^{2} + x_{2}^{2} = N

is expressed as

NS \approx \frac{cN}{lnN}

(43)

where

c

is a constant,

c = \frac{\sum_{i = 1}^{k} (\frac{1}{lnk})}{2 \sum_{i = 1}^{2 k} (\frac{1}{lnk})} > \frac{1}{4}

According to lemma 3, when

m = 3

, the standard Diophantine equation.

y_{1} + y_{2} + y_{3} = N

(44)

The approximate number of integer solutions is

NS \approx c \frac{1}{N} |y_{1}| |y_{2}| |y_{3}|

(45)

In the equation

p + x_{1}^{2} + x_{2}^{2} = N

, the solution set is

y_{1} = p = \{2, 3, 5, \dots, N\}

{y_{2} = y}_{3} = x_{1}^{2} = x_{2}^{2} = \{1^{2}, 2^{2}, 3^{2}, \dots, j^{2}, \dots, N\}

, the number of solution sets

|y_{1}| = |p| = \frac{N}{lnN}

|y_{2}| = |y_{3}| = |x_{1}^{2}| = {|x_{2}^{2}| = N}^{\frac{1}{2}}

The approximate number of integer solutions (NS) of the equation

p + x_{1}^{2} + x_{2}^{2} = N

is expressed as

NS \approx c \frac{1}{N} \frac{N}{lnN} N^{\frac{1}{2}} N^{\frac{1}{2}} = c \frac{N}{lnN}

(46)

where

c

is a constant,

c = \frac{\sum_{i = 1}^{k} (\frac{1}{lnk})}{2 \sum_{i = 1}^{2 k} (\frac{1}{lnk})} > \frac{1}{4}

When

N

is odd, the number of solution sets

|x_{1}| = \frac{N}{lnN}

|x_{2}| = |x_{3}| = \frac{N^{\frac{1}{2}}}{2}

The approximate number of integer solutions (NS) of the equation

p + x_{1}^{2} + x_{2}^{2} = N

is expressed as

NS \approx c \frac{1}{N} \frac{N}{lnN} \frac{N^{\frac{1}{2}}}{2} \frac{N^{\frac{1}{2}}}{2} = c \frac{N}{lnN}

(47)

The proof is completed.

Nevanlinna theory deals with the growth of zeros and poles (roots or solution sets) of meromorphic functions. According to Vojta's conjecture, there is a corresponding (similar) relationship between Nevanlinna theory in function fields and the abc conjecture in number fields (including integers)

[12-14]

. Since the general Fermat conjecture

[9-11]

can be derived from the abc conjecture and also concerns the number of zeros (roots or solutions) of equations, Nevanlinna theory, the abc conjecture, Vojta's conjecture, and the general Fermat conjecture all of those problem is the same class of problems related to the number of zeros (roots or solutions) of equations.

This article presents an approximate combinatorial method for estimating the number of integer solutions to general Diophantine equations with countable solution sets. The applicable countable solution sets include integers, rational numbers, prime numbers, algebraic numbers, polynomials, algebraic number fields, and function fields. Some problems can also be transformed into counting number of solutions of equations

[15]

. For applications of this method in other areas such as the essence and number of solutions of the abc conjecture, the number of solutions to the general Fermat conjecture, and so forth, please refer to the reference

[6]

Abbreviations

NS	Number of Solutions

Author Contributions

Shihui You: Conceptualization, Formal Analysis, Investigation, Methodology

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	C. Hooley. On the representation of a number as the sum of two squares and a prime. Acta Math. 97 (1957), 189-210. https://doi.org/10.1007/BF02392397
[2]	G. H. Hardy, J. E. Littlewood. Some problems of partitio numerorum; III: on the expression of a number as a sum of primes. Acta Math. 44 (1923), 1-70. https://doi.org/10.1007/BF02403921
[3]	Ricardo Adonis Caraccioli Abrego. Every sufficiently large odd integer is a sum of two positive perfect squares and a prime. 2025. https://arxiv.org/abs/2509.14246
[4]	Ju. V. Linnik. An asymptotic formula in an additive problem of Hardy-Littlewood. Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 629-706. https://www.mathnet.ru/eng/im/v24/i5/p629
[5]	C. Hooley. Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics 70, Cambridge University Press 1976.
[6]	Shihui You. On the number of solutions of the equation with countable solution set. https://www.researchgate.net/publication/391160710
[7]	Takao Komatsu. On the number of solutions of the Diophantine equation of Frobenius-General case. Mathematical Communications 8 (2003), 195-206. https://hrcak.srce.hr/file/1447
[8]	Shihui You. Enumeration method to compute the number of solutions of integer to the linear Diophantine equations. June 2025. https://www.researchgate.net/publication/392561561
[9]	R. Mauldin. A generalization of Fermat’s Last Theorem: The Beal conjecture and prize problem. Notices Am. Math. Soc. 44 (1997), no. 11, p 1436–1437. https://www.ams.org/notices/199711/beal.pdf
[10]	Shihui You. Number of solutions of general Fermat equation. April 2025. https://www.researchgate.net/publication/391160858
[11]	Shihui You. A Short Proof of Fermat Theorem and General Fermat (or Beal) Conjecture. April 2025. https://www.researchgate.net/publication/391219339
[12]	Michel Waldschmidt. Lecture on the abc conjecture and some of its consequences. 6th World Conference on 21st Century Mathematics 2013. (P. Cartier, A. D. R. Choudary, M. Waldschmidt Editors), Springer Proceedings in Mathematics and Statistics 98 (2015), 211-230. https://doi.org/10.1007/978-3-0348-0859-0_13
[13]	Shihui You. Essence of abc conjecture of integer. April 2025. https://www.researchgate.net/publication/391160242
[14]	Shihui You. A Short Proof of abc Conjecture of Integer. April 2025. https://www.researchgate.net/publication/391219998
[15]	Shihui You. On Erdos and Turan conjecture on arithmetic progressions. May 2025. https://www.researchgate.net/publication/391353598

Cite This Article

Plain Text BibTeX RIS

APA Style

You, S. (2026). On the Representation of a Number as the Sum of Two Squares and a Prime. Pure and Applied Mathematics Journal, 15(2), 18-28. https://doi.org/10.11648/j.pamj.20261502.12

Copy | Download

ACS Style

You, S. On the Representation of a Number as the Sum of Two Squares and a Prime. Pure Appl. Math. J. 2026, 15(2), 18-28. doi: 10.11648/j.pamj.20261502.12

Copy | Download

AMA Style

You S. On the Representation of a Number as the Sum of Two Squares and a Prime. Pure Appl Math J. 2026;15(2):18-28. doi: 10.11648/j.pamj.20261502.12

Copy | Download

@article{10.11648/j.pamj.20261502.12,
  author = {Shihui You},
  title = {On the Representation of a Number as the Sum of Two Squares and a Prime},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {18-28},
  doi = {10.11648/j.pamj.20261502.12},
  url = {https://doi.org/10.11648/j.pamj.20261502.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.12},
  abstract = {This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - On the Representation of a Number as the Sum of Two Squares and a Prime
AU  - Shihui You
Y1  - 2026/04/20
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.12
DO  - 10.11648/j.pamj.20261502.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 18
EP  - 28
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.12
AB  - This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.
VL  - 15
IS  - 2
ER  -

Copy | Download

Author Information

Shihui You

School of Intelligent Manufacturing Engineering, Linyi University of Technology, Linyi, China

Biography: Shihui You is a professor at Linyi University of Technology in Intelligent Manufacturing, completed PhD in Material Process Engineering from South China University of Technology in 2008, and master’s in mechanical engineering from Xiangtan University in 1990. Professor, director of the key laboratory of colleges and universities in Hunan province, member of the academic committee of Xiangtan University, master, doctor and postdoctoral supervisor in Xiangtan University in 2010-2018. Professor, member of the academic committee of Jiujiang University, Vice president of School of Mechanical and Material Engineering of Jiujiang University, Director of the provincial key laboratory of Jiujiang University in 2003-2010. Presided over and completed one international research collaboration projects of the National Ministry of Science and Technology of China (with University of Glasgow) and one general project of Natural Science Foundation in China. published more than 70 papers at home and abroad.

Research Fields: Number Theory, Combinatorics, Graph Theory, Geometry and Topology, Artificial Intelligence, Mechanics, Mechanical Engineering, Civil Engineering, Materials Engineering, Naval Architecture

Contact Email

http://orcid.org/0009-0008-1565-7669

Download PDF

Submit an Article

Plain Text BibTeX RIS

APA Style

You, S. (2026). On the Representation of a Number as the Sum of Two Squares and a Prime. Pure and Applied Mathematics Journal, 15(2), 18-28. https://doi.org/10.11648/j.pamj.20261502.12

Copy | Download

ACS Style

You, S. On the Representation of a Number as the Sum of Two Squares and a Prime. Pure Appl. Math. J. 2026, 15(2), 18-28. doi: 10.11648/j.pamj.20261502.12

Copy | Download

AMA Style

You S. On the Representation of a Number as the Sum of Two Squares and a Prime. Pure Appl Math J. 2026;15(2):18-28. doi: 10.11648/j.pamj.20261502.12

Copy | Download

@article{10.11648/j.pamj.20261502.12,
  author = {Shihui You},
  title = {On the Representation of a Number as the Sum of Two Squares and a Prime},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {18-28},
  doi = {10.11648/j.pamj.20261502.12},
  url = {https://doi.org/10.11648/j.pamj.20261502.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.12},
  abstract = {This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - On the Representation of a Number as the Sum of Two Squares and a Prime
AU  - Shihui You
Y1  - 2026/04/20
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.12
DO  - 10.11648/j.pamj.20261502.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 18
EP  - 28
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.12
AB  - This paper shows that every integer and odd integer are a sum of a prime and two squares. We solved this by transforming the problem of the number of such representations into finding the number of integer solutions to the corresponding Diophantine equation. To find integer solutions to nonlinear or higher-order Diophantine equations is very challenging, whereas counting the number of integer solutions for the linear Diophantine equations is comparatively easier. The methods involve the combinatorial approximation method to obtain the number of integer solutions of the Diophantine equation with the countable solution sets. First, the number of integer solutions to the linear Diophantine equations with integer sets or sequences as solution sets can be determined by combinatorial methods. Second, the approximate number of integer solutions of the linear Diophantine equations with integer sets or sequences as solution sets is obtained using the approximate method. Finally, for cases where the solution sets and the number of solution sets of the variables in the linear Diophantine equations are either the same or different, we propose the approximate combinatorial method to compute the approximate number of integer solutions of the linear Diophantine equations whose solution sets are integer subsets or sequences of integer subsets. The results are consistent with the existing conclusions. The purpose of this paper is to verify the adaptability and correctness of the approximate combinatorial methods for solving the number of integer solutions of the general Diophantine equations with countable solution sets.
VL  - 15
IS  - 2
ER  -

Copy | Download