1. Introduction

Diophantine equation, also called indefinite equation, is named by the ancient Greek mathematician Diophantus. A Diophantine equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are also integers [1]. The research on a Diophantine equation often base on three problems: 1. This equation has integral solutions in what conditions. 2. If the equation does have integral solutions, how many integral solutions are there? 3. Obtain the general solution. It is hard to determine whether a Diophantine equation has integral solutions or not and find the general solution without any specific method. The common methods are quadratic residue, congruence, Diophantine approximation, method of infinite descent and so on [2]. Pell equation is a kind of Diophantine equation, initially in the form of x²-Dy²=1, which is first studied by Fermat, who mentioned the equation x²-Dy²=1 again where D is a positive integer that is not a perfect square [3]. In the following paragraphs, this paper expands the theories of Pell equation and summarizes the theorems provided by some documents and the key points in their proofs about the integral solutions of the equations in the form of xª±b=Dy²where a=2 or a=3. The solutions of these equations not only promote the development of number theory, but also become the basis of a practical mathematical model that contribute to the application in many fields such as computer science and cryptography.

2. On the equation x²±b=Dy²

2.1. Theorems about x²+1=Dy² and x²-4=Dy²

In the work by Chen in 2020[4], according to the application of Störmer theorem on Pell equations and the utilization of theories of Lehmer sequence, for the equation x²+1=Dy², the relation between a group of positive integral solution (x,y) and its fundamental solution if there is specifically one or two prime divisors of y not dividing D is obtained, for the equation x²-4=Dy², the relation between a group of positive integral solution (x,y) and its fundamental solution if there is exactly two prime divisors of y not dividing D is obtained. As a relevant theorem, the Störmer theorem states:

letting (x₁,y₁) be a group of positive integer solution of the equation x²-Dy²=±1, if all the prime divisors of y₁ divides D, then x₁+y₁ \( \sqrt[]{D} \) is the fundamental solution of the equation x²-Dy²=±1.

Chen mentioned and proved three theorems in this work, now we just consider one of them, which states:

letting (x,y) be a group of positive integer solution of the equation x²+1=Dy², y=pⁿy’, p ∤D is a prime number, D is not a perfect square, n∈N. If any prime divisor of y’ divides D, then x+y \( \sqrt[]{D} \) =ℰ or ℰq, where q is an odd prime and p ≠q. x+y \( \sqrt[]{D} \) =ℰ is the fundamental solution of the equation x²+1=Dy²[5].

In the proof of this theorem, Chen used the concept of congruence, Legendre symbol and quadratic residue. By Chen’s theorems, all the equations consistent with the conditions can be determined as long as their fundamental solutions are found.

2.2. Theorems about x²-1=15y²and y²-1=Dz²

Pu and Wan discussed the common solution of the equations x²-1=15y²and y²-1=Dz²in 2017[6]. Some relevant lemmas are:

Letting D>0 is not a perfect square, the fundamental solution of the Pell equation x⁴-Dy²=1 is (x0,y0), then the equation x⁴-Dy²=1 has a solution if and only if x₀ or 2x₀²-1 is a perfect square.

2. When a>0 is a perfect square, the equation ax⁴-by²=1 has at most one group of positive integer solution.

3. If D is a non-square positive integer, then the equation x²-Dy⁴=1 has at most one group of positive integer solution (x,y), also the equation has two groups of positive integer solution if and only if D=1785 or D=28560 or 2x₀ and y₀ are both perfect squares, where (x₀,y₀) is the fundamental solution of the equation x²-Dy²=1.

4. (I) The only positive integer solution of the equation x⁴-15y²=1 is (x,y)=(2,1).

(II) The only positive integer solution of the equation 16x⁴-15y²=1 is (x,y)=(1,1).

(III) The only positive integer solution of the equation x²-15y⁴=1 is (x,y)=(4,1).

5. Letting all the integral solutions of the equation x²-15y²=1 be (x_n,y_n), n ∈Z, then

(I) x_n is a perfect square if and only if n=0 or n=1.

(II) \( \frac{{x_{n}}}{4} \) is a perfect square if and only if n=1 or n=-1.

(III) yn is a perfect square if and only if n=0 or n=1.

The theorem provided by Pu and Wan suggests:

if D=2p₁…p_s (1≤s≤3), p₁,…,p_s (1≤s≤3) are distinct odd prime numbers, then the Pell equations x²-1=15y²and y²-1=Dz²have non-trivial solution (x,y,z)=(±244,±63,±8) and trivial solution (x,y,z)=(±4,±1,0) when D=2 \( × \) 33. When D=2 \( × \) 7 \( × \) 31 \( × \) 61, the equations have non-trivial solution (x,y,z)=(±15124,±3905,±24) and trivial solution (x,y,z)=(±4,±1,0).

To prove this theorem, letting (x,y,z)=(xn,yn,,z) where n∈Z be the integer solution of the equations x²-1=15y²and y²-1=Dz². The conditions of different parities of n have to be considered respectively. The principally methods Pu and Wan used are considering some natures of odd and even numbers, recursive sequence and congruence.

2.3. Theorems about x²-1=6y²and y²-4=Dz²

The study by Guan [7] gave two important theorems:

If p₁,…,p_s are distinct odd prime numbers, D=p₁…p_s (1≤s≤3), then the simultaneous equations x²-1=6y²and y²-4=Dz²where x,y,z∈Z

have non-trivial solution (x,y,z)=(±49,±20,±6) and trivial solution (x,y,z)=(±5,±2,0) when D=11

have non-trivial solution (x,y,z)=(±4801,±1960,±6) and trivial solution (x,y,z)=(±5,±2,0) when D=11 \( × \) 89 \( × \) 109

have non-trivial solution (x,y,z)=(±4656965,±1901198,±840) and trivial solution (x,y,z)=(±5,±2,0) when D=11 \( × \) 97 \( × \) 4801

only have trivial solution (x,y,z)=(±5,±2,0) when D \( ≠ \) 11,11 \( × \) 89 \( × \) 109,11 \( × \) 97 \( × \) 4801.

When D is even, if D has no prime divisor p which satisfies p \( ≡ \) 1 mod 24 and p \( ≡ \) 7 mod 24, then the simultaneous equations x²-1=6y²and y²-4=Dz²have the trivial solution (x,y,z)=(±5,±2,0) only.

To prove the first theorem, letting (±xm,±ym,,±z) be all of the solutions of the simultaneous equations x²-1=6y²and y²-4=Dz², where ym and z must be even. There are two situations required to be discussed that 2 | m or 2 ∤m. To justify the two theorems, the writer mainly considered the parity of D in different conditions. What is more, in the demonstration of the second theorem, the writer converted the equations to x²-24a²=1 and a²-Db²=1 where x,a,b∈N∗. It is viable to do so since y and z can be expressed as y=2a, z=2b where a,b∈N∗and D∈N∗has no square divisor and D \( ≡ \) 2 mod 4 when D is even.

3. On the equation x³±b=Dy²

3.1. Theorems about x³±1=Dy²

For Diophantine equations x³-1=Dy²and x³+1=Dy² where D>2 and cannot be divided by 3 or other prime numbers in the form of 6k+1. In 1942, Ljunggren attested that the equations x³-1=Dy²and x³+1=Dy² have at most one group of positive integer solution (x,y)[8]. In 1981[9], Ke and Sun demonstrated one theorem respectively for the equation x³-1=Dy²and x³+1=Dy2 that the only integral solution of the equation x³-1=Dy²is (x,y)=(1,0) and the only integral solution of the equation x³+1=Dy²is (x,y)=(-1,0), by using elementary methods only. For the equation x³-1=Dy², first of all the authors utilized the method of factorization to express the equation as (x-1)(x²+x+1)=Dy². It is obviously that the greatest common divisor of (x-1) and (x²+x+1) can be 1 or 3 only, then the two situations that gcd(x-1,x²+x+1)=1 or gcd(x-1,x²+x+1)=3 were considered severally. After a few steps, the situation that gcd(x-1,x²+x+1)=1 was shown to be impossible. Therefore, as gcd(x-1,x²+x+1)=3, x-1, x²+x+1 and y can be expressed in the form of u and v (u>0, v>0) that x-1=3Du², x²+x+1=3v², y=3uv. In the following manipulations, x=3Du²+1 was substituted to x²+x+1=3v², a new equation was then obtained. Two conditions whether D is divisible by 2 also have to be discussed respectively since it is related to two different expressions of u. The highlight of the study by Ke and Sun is that they used elementary methods such as factorization and quadratic residue only.

3.2. Theorems about x³±27=Dy²

In 1988, in order to add the work by Ke and Sun in 1981[9], Cao provided a new theorem severally for the equation x³-1=3y²and x³±1=6y²that the only integral solution for x³-1=3y² is the trivial solution (x,y)=(1,0) and the only integral solution for x³±1=6y² is the trivial solution (x,y)=(±1,0)[10]. Cao also used the method of factorization to turn x³-1=3y² into (x-1)(x²+x+1)=3y², then expressed x-1, x²+x+1 and y as x-1=9a², x²+x+1=3b², y=3ab, substituted x to x²+x+1 for the following manipulations. Before the theorem about x³±27=Dy², Cao stated two prerequisite theorems:

For the equation x³+1=3Dy² where D>0, D=1 or D has no square divisor and cannot be divisible by the prime numbers in the form of 6k+1, the only integral solution of the equation x³+1=3Dy² is the trivial solution (x,y)=(-1,0).

For the equation x³-1=Dy² where D>0, D=1 or D has no square divisor and cannot be divisible by the prime numbers in the form of 6k+1, the only integral solution of the equation x³-1=Dy² is the trivial solution (x,y)=(1,0).

By extending these two theorems, Chen testified that for the equation x³+27=Dy² where D>0 and D has no prime odd power divisor in the form of 6k+1, except for some values of D that the equation does have at most two non-trivial integral solutions, the equation has trivial solution (x,y)=(-3,0) only. To prove this theorem, the lemmas that the equation x³+1=y² has integral solutions (0,±1) and (2,±3) only besides the trivial solution (-1,0) and the equation x³+1=2y² has integral solutions (1,±1) and (23,±78) only besides the trivial solution (-1,0) are needed. Another theorem provided by Cao about the equation x³-27=Dy² suggests that for the equation x³-27=Dy² where D>0 and D has no prime odd power divisor in the form of 6k+1, except for the situation when D=2 the equation has solution (5,±7) only and the situation when D=98 the equation has solution (5,±1) only, the only solution of the equation x³-27=Dy² is the trivial solution (3,0). To prove this theorem, the two conditions 3 | x or 3 ∤x have to be considered.

3.3. Another way to solve the equation x³±27=Dy² and the theorems about x³±729=Dy²

Founded on the study on the equation x³±1=Dy²by Ke in 1988[9], Si and Pang provided all the non-trivial solutions of the equation x³±27=Dy²(D>0, has no divisor of perfect square and prime numbers in the form of 6k+1) in another way and all the non-trivial solutions of the equation x³±729=Dy²[11]. Lemmas:

1. For the equation x³+1=Dy²,

(I) when D is not the multiple of 2 or 3, the integral solutions are (-1,0), (0,1), (2,3).

(II) when D is a multiple of 2, the integral solutions are (-1,0), (0,1), (23,78).

(III) when D is a multiple of 3, the integral solution is (-1,0).

2. For the equation x³-1=Dy2, the only integral solution is (1,0).

In theorems by Si and Pang, the non-trivial solutions of the equation x³+27=Dy²when D=3, 6, 11, the only non-trivial solution of the equation x³-27=Dy²when D=2, the non-trivial solutions of the equation x³+729=Dy²when D=33, 1, 2, 74, the non-trivial solutions of the equation x³-729=Dy²when D=47, 6 are given. To prove the theorems above, the situations of 3 | x or 3 ∤x have to be considered.

4. Conclusion

This paper discussed mainly on the Diophantine equation xª±b=Dy²where a=2 or a=3. All of the theorems above consummate our study on the Diophantine equation. However, the researches on the Diophantine equations are so broad that the content of this paper is only the tip of the iceberg. Even one particular equation can be solved in many different ways, the methods mentioned in this paper can also be used to solve other equations. The research orientation in the future is to use known methods to solve equations of other values of b and D, also the discovery of new methods.