SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v16i41.2338research articleExistence and Non - Existence of Exponential Diophantine Triangles Over Triangular NumbersMahalakshmiM1KannanJjayram.kannan@gmail.com2DeepshikaA1KaleeswariK2Research Scholar, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to Madurai Kamaraj University)Sivakasi, Tamil NaduIndiaAssistant Professor, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to Madurai Kamaraj University)Sivakasi, Tamil NaduIndia3110202316413599149202324920232023
<bold id="s-130efd7cdb38">Abstract</bold>

Objectives: The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers (tn,n∈ℕ). Methods: An Exponential Diophantine triangle over triangular numbers (tn,n∈ℕ) is defined as a triangle with sides nx+1,ny+2 and nz where x,y, and z are non - negative integers such that tnx+tn+1y=z2. To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial expansion, and various theories concerning congruence are employed. Findings: Here it is proved that, for five different choices of sides, an Exponential Diophantine triangle over tn can be constructed. In particular, infinitely many such triangles can be found. For some particular choice of sides, Python coding is displayed along with its output to verify the existence of required triangles. On the other side, another five different choices of sides are considered and it is shown that no considered type of triangles exists in these cases. Novelty: The idea of solving an exponential Diophantine equation and the idea of constructing triangles under some conditions using Diophantine equations already exists in the mathematical society. This article is created uniquely by combining these two concepts along with the innovative usage of exponential Diophantine equations.

In this article, an Exponential Diophantine triangle over triangular numbers (tn) is defined as a triangle with sides nx+1,ny+2 and nz where x,y, and z are non-negative integers such that tnx+tn+1y=z2. Hereafter, throughout this paper, such triangles are denoted as ED triangles over tn. The research problem taken for study is the existenceand non-existence of such triangles. That is, here are the situations under which an ED triangle over tn exists and the situations under those triangles not exist are discussed.

With a deep point of view, one can observe that the research problem considered here depends mainly on solving the exponential Diophantine equation over non-negative integers. This article is motivated by the works1, 2 and can be considered as next-level research in this area.

Previous works of researchers involve solving the exponential Diophantine equations alone or construct a geometrical shape with some special conditions through Diophantine equations such as Pythagorean equations, Pell equations3, 4, 5, 6, 7, 8, 9, 10, 11, 12. This work connects both the ideas and it is undertaken to move this type of research to a somewhat different extent.

Let us discuss some key terms used in this article. First of all, the dominating factor in this work is the study of exponential Diophantine equation tnx+tn+1y=z2 (An exponential Diophantine equation is a type of Diophantine equation in which the exponents are known or unknown variables). To obtain the existence and non-existence, one has to solve this exponential Diophantine equation. Apart from using the usual Catalan’s conjecture, basic properties of congruence, we employ certain factors in addition to solve this equation. Such factors include binomial expansion and negative Pell’s equation.

Excluding introduction, conclusion, and references, this article comprises three different sections: Methodology (section 2), Existence of ED triangles over tn (subsection 3.1) and non-existence of ED triangles over tn (subsection 3.2). In each of these sections, the article successfully attained its aim.

<bold id="s-29c642ba2cbf">Methodology</bold>

This section includes the necessary preliminary things needed for the main work of this article. It includes Catalan’s conjecture, Negative Pell’s equation, and some elementary results.

Conjecture 2.1.(3,2,2,3)is a unique solution (a,b,x,y) for the Diophantine equation ax-by=1, where a,b,x and y are integers such that mina,b,x,y>1^{12}.

This conjecture is known as Catalan’s conjecture.

Definition 2.2. The equation x2-dy2=-1 is called the Negative Pell’s equation where d∈ℕ and it is not a perfect square.

Definition 2.3. The equation x2-dy2=-n is called the Generalized Negative Pell’s equation where d,d∈ℕ and dis not a perfect square.

Lemma 2.4. If l and k are positive integer solutions of the generalized negative Pell’s equation l2-8k2=-7, then the values of l and k (say lr&kr,r=1,2,3,⋯) can be obtained by the relations

lr=6lr-2-lr-4kr=6kr-2-kr-4

for r≥5, provided l1=1,l2=5,l3=11,l4=31,k1=1,k2=2,k3=4 and k4=11.

Proof. We aim to show that Equation 1 satisfies the relation lr2-8kr2=-7 for all r≥5. Using Equation 1, one can write

The result follows once Equation 3 is substituted in Equation 2.

Lemma 2.5. For n\in N and n>1, n^4+2n^3+3n^2+6n+4 is not a perfect square.

Proof. Suppose that\;n^4+2n^3+3n^2+6n+4={\left(n^2+an+b\right)}^2. Then comparing the coefficient of n^3 and the constant term, we get b^2=4,\;2a=2. These choices imply b=2,\;a=1. This is a possible choice of a and b. Also, we need to equate the terms 3n^2+6n and \left(a^2+2b\right)n^2+2abn. If we do so, we obtain n=\frac{2(ab-3)}{3-a^2-2b}. Since n \in \mathbb{N}, we have two possibilities:

i) ab>3 and a^2+2b<3

ii) ab<3 and a^2+2b>3

For the case (i), we could not have any suitable a and b. But for case (ii), we have two suitable a and b, as a=1,b=2(which was the earlier one) and a=2,b=1(an impossible one). Thus, the only choice of a and b is a=1 and b=2. This leads to the value of n as\;1. This completes the proof.

Lemma 2.6. For n \in \mathbb{N} and n>2, n^4+4n^3+11n^2+2n+4 is not a perfect square.

Proof. The proof is similar to that of Lemma 2.5.

Definition 2.7. For n \in \mathbb{N}, a triangular number t_n is defined as t_n=\frac{n(n+1)}2.

Lemma 2.8. The sum of two consecutive triangular numbers is a perfect square. In particular, t_n+t_{n+1}={\left(n+1\right)}^2.

These are the methods or results we employ in this paper to solve the exponential Diophantine equation.

<bold id="s-7361fb3fe13a">Results and Discussion</bold>

This section is split into two subsections 3.1 and 3.2. In subsection 3.1, it is discussed and proved that the ED triangles over t_n exist for the following sides:

• 1,n+2 and nz

• n+1,\;2 and nz

• n+1,\;n+2 and nz

• x+1,\;3, and z(x>1)

• nx+1,ny+2, and nz(x,y>1)

In subsection 3.2, it is discussed and proved that the ED triangles over t_n does not exist for the following sides:

• 1,\;2, and nz

• 1,\;ny+2 and nz(y>1)

• nx+1,\;2 and nz(x>1)

• nx+1,\;n+2 and nz(x>1)

• n+1,ny+2\; and nz(y>1)

<bold id="s-1e5a4d9e882e">3.1 Existence of ED Triangles over <italic id="e-e195013c260e">t<sub id="s-b5862919b884">n</sub></italic></bold>

In this subsection, the existence of ED triangles over t_n with some particular sides are discussed. Also, for the choice x>1,y>1, a python code is displayed to determine the existence.

Theorem 3.1.1. There exist infinitely many ED triangles over t_n with sides 1,n+2 and nz for some n\in N and z\in N\cup\{0\}.

Proof. Comparing the sides 1,n+2,\;and nz with the sides nx+1,ny+2, and nz, we see that x=0 and y=1. Substituting these values of x and y in t_n^x+t_{n+1}^y=z^2, we obtain 1+t_{n+1}=z^2. Replacing t_{n+1} by \frac{(n+1)(n+2)}2, we receive that z=\sqrt{\frac{n^2+3n+4}2}. Since z is an integer, we must have that n^2+3n+4={2k}^2 for some k \in \mathbb{Z}. Considering this as a quadratic equation in n, the possible value of n is obtained as n=\frac{-3+\sqrt{8k^2-7}}2. For this n to be a natural number, one must have that 8k^2-7=l^2 for some l \in \mathbb{Z}. By Lemma 2.4, there are infinitely many such l and k's exist. So we found infinitely many n (say n_r) which is calculated as \frac{l_r-3}2(r\geq1).

Theorem 3.1.2. There exist infinitely many ED triangles over t_n with sides n+1,\;2 and nz for some n\in\mathbb{N},\;\mathrm{and}\;z\in\mathbb{N}\cup\{0\}.

Proof. Comparing the sides n+1,\;2,\;and nz with the sides nx+1,ny+2, and nz, we see that x=1 and y=0. Substituting these values of x and y in t_n^x+t_{n+1}^y=z^2, we obtain t_n+1=z^2. Doing the same process as in Theorem 3.1.1, it is seen that n_r=\frac{l_r-1}2\left(r\geq1\right).

Theorem 3.1.3. There exist infinitely many ED triangles over t_n with sides n+1,\;n+2 and nz for some n\in\mathbb{N},\;\mathrm{and}\;z\in\mathbb{N}\cup\{0\}.

Proof. Here x=y=1. So the obtained equation is t_n+t_{n+1}=z^2. By Lemma 2.8, the result follows.

Corollary 3.1.4. For any natural number n, one can construct an ED triangle over t_n.

Theorem 3.1.5. There exist infinitely many ED triangles over t_n with sides x+1,\;3, and z for some x\left(>1\right),\;z\in\mathbb{N}\cup\{0\}.

Proof. Here n=1,y=1. So the obtained equation is 1^x+3=z^2\;. For any choice of x>1, this equation is satisfied.

Theorem 3.1.6. There exist ED triangles over t_n with sides nx+1,ny+2, and nz for some x\left(>1\right),y\left(>1\right),\;z\in N\cup\{0\}.

The existence of this case is given by the following Python coding (Figure 1) and its output (Figure 2).

<bold id="strong-d366a33703b141eda83b99ce1ffeeb53">Python Coding for the existence of ED triangles over </bold>
<inline-formula id="inline-formula-168544d692284ac7b249d6580ce2461b"> <tex-math>t_n</tex-math></inline-formula>
<bold id="strong-c4ddf98895b94e22a6ecb24594c9167f"> for </bold>
<inline-formula id="inline-formula-e927ab9ac0b44655a153b0189dd3a149"> <tex-math>x,y>1</tex-math></inline-formula>
<bold id="strong-788aefbb281c4a80b5a4e8dbc9cfe55a">Output for the coding in <xref id="x-85361687849d" rid="figure-58c8c83c4e8249e88b71d46f431fe38d" ref-type="fig">Figure 1</xref></bold>
<bold id="s-45b61ccac4ec">3.2 Non - Existence of ED Triangles over <italic id="e-58696818508d">t<sub id="s-05bd4c20f007">n</sub></italic></bold>

In this section, the non-existence of ED triangles over t_n with some particular sides discussed using Catalan’s conjecture and some properties of congruence ^{13}.

Theorem 3.2.1. One cannot find an ED triangle over t_n with sides 1,\;2, and nz for some n\in\mathbb{N},\;\mathrm{and}\;z\in\mathbb{N}\cup\{0\}.

Proof. Comparing the sides 1,2\;and nz with the sides nx+1,ny+2, and nz, we see that x=0 and y=0. Incorporating these values of x and y in t_n^x+t_{n+1}^y=z^2, the latter equation becomes z^2=2, which is an impossible one.

Theorem 3.2.2. One cannot find an ED triangle over t_n with sides 1,\;ny+2 and nz for some n\in\mathbb{N},\;\mathrm{and}\;z\in\mathbb{N}\cup\{0\}.

Proof. Comparing the sides 1,ny+2,\;and nz with the sides nx+1,ny+2, and nz, we see that x=0. Incorporating this value of x in t_n^x+t_{n+1}^y=z^2, this exponential Diophantine equation becomes z^2-t_{n+1}^y=1. By Catalan’s conjecture, t_{n+1}=2. But 2 is not a triangular number.

Theorem 3.2.3. There are no ED triangles over t_n exists with sides nx+1,\;2 and nz for some n\in\mathbb{N},\;\mathrm{and}\;z\in\mathbb{N}\cup\{0\}.

Proof. By comparing the sides as usual, here it is noted that y=0. If so, we get the exponential Diophantine equation as z^2-t_n^x=1. If n>1, then by Catalan’s conjecture, t_n=2, which is not possible. If n=1, then z^2=2. This is also not possible.

Theorem 3.2.4. There are no ED triangles over t_n exists with sides nx+1,\;n+2 and nz for some n \in \mathbb{N} \backslash\{1\} and x(>1),z \in \mathbb{N} \cup\{0\}.

Proof. Here y=1. Since n>1, we must have that t_n>1 and so t_n^x>t_n. This leads to the fact that z^2>t_n^x+t_{n+1}>t_n+t_{n+1}. Applying Lemma 2.8, it is clear that z^2>{\left(n+1\right)}^2. So take z=n+k for some k=2,3,\cdots. Thus, the equation t_n^x+t_{n+1}=z^2 becomes

If x=2, then Equation 4 becomes 4k^2+8nk+\left(-n^4-2n^3+n^2-6n-4\right)=0. Solving this for k, we arrive at k=\frac{-8n\pm4\sqrt{n^4+2n^3+3n^2+6n+4}}8. Using Lemma 2.5, we conclude that k is not an integer.

Suppose x>2. Then comparing the coefficients of n^2 in Equation 4 gives 2^{x-1}-2^x=0. This leads to an impossible situation that -1=0.

Theorem 3.2.5. There are no ED triangles over t_n exists with sides n+1,ny+2\; and nz for some n \in \mathbb{N} and y(>1), z \in \mathbb{N} \cup\{0\}.

Proof. Here x=1. If n=1, then the considered exponential Diophantine equation becomes 1+3^y=z^2. This has no solution byCatalan’s conjecture. If n=2, then we’ve 3+6^y=z^2. Since y>1, 6^y\equiv0\;(mod\;4). This leads to z^2\equiv3\;(mod\;4). This is not possible. Assume that n>2. Then as in Theorem 3.2.4, one can take z=n+k for some k=2,3,\cdots and by the same process we arrive at the relation

If y=2, then Equation 5 becomes 4k^2+8nk-n^4-4n^3-7n^2-2n-4=0. By Lemma 2.6, we conclude that k is not an integer.

Suppose y>2. Then equating the coefficients of n^2 in Equation 5 gives

2^{y-1}-2^y+2^{y-2}\begin{pmatrix}y\\y\;-\;2\end{pmatrix}+\begin{pmatrix}y\\y\;-\;1\end{pmatrix}2^{y-1}\begin{pmatrix}y\\1\end{pmatrix}+\begin{pmatrix}y\\2\end{pmatrix}2^y=0y=\frac59\not\in Z

This completes the proof.

<bold id="s-cac4484eb81c">Conclusion</bold>

In this article, we successfully discussed the existence and non-existence of Exponential Diophantine triangles over t_n. Apart from just existence, the exact sides of the triangles are mentioned. Also, Python coding to verify the existence case for the choice x,y>1 is provided. This article stands in a new path in the sense of employing an exponential Diophantine equation to construct a newfangled set of triangles. Also, with a deep point of view, one can understand that all the possible choices of sides are discussed thoroughly. This research can be extended, modified or generalized in many ways. Some of these include replacing triangular numbers with some other special numbers or changing the form of exponential Diophantine equations.

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