# On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

- (i)
- There uniquely exist rational polynomials $B\left(x\right)$ and $C\left(x\right)$ with $deg\left(C\left(x\right)\right)\le \frac{k-1}{2}$ such that$${f}_{k}\left(x\right)={B}^{2}\left(x\right)+C\left(x\right).$$
- (ii)
- Let l be the least positive integer such that $lB\left(x\right)$ and ${l}^{2}C\left(x\right)$ have integer coefficients for any nonnegative integer i and $\delta \in \{1,-1\}$$${P}_{i,\delta}\left(x\right)=\delta {(lB\left(x\right)+\delta i)}^{2}-\delta {\left(lB\left(x\right)\right)}^{2}-\delta {l}^{2}C\left(x\right),$$$${H}_{1}=\{\alpha \in \mathbb{Z}:{P}_{i,\delta}\left(\alpha \right)=0,\delta \in \{1,-1\},i=0,1,2,\dots ,r-1\},$$$${H}_{2}=\{\alpha \in \mathbb{R}:{P}_{r,1}\left(\alpha \right)=0\phantom{\rule{3.61371pt}{0ex}}\mathrm{or}\phantom{\rule{3.61371pt}{0ex}}{P}_{r,-1}\left(\alpha \right)=0\},$$$$min{H}_{2}\le x\le max{H}_{2}.$$

## 2. Proofs

**Lemma**

**1.**

**Proof.**

**Lemma**

**2**

**Lemma**

**3**

**Lemma**

**4**

**Lemma**

**5**

**Proof of Theorem 1.**

**Proof of Corollary 1.**

**Proof of Theorem 2.**

**Note 1.**

**Note 2.**

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Shorey, T.N.; Tijdeman, R. Exponential Diophantine Equations; Cambridge University Press: Cambridge, UK, 1986. [Google Scholar]
- Tijdeman, R. Applications of the Gel’fond-Baker method to rational number theory. In Topics in Number Theory, Proceedings of the Conference Debrecen 1974; Colloquia Mathematica Societatis Janos Bolyai; North-Holland: Amsterdam, The Netherlands, 1976; Volume 13, pp. 399–416. [Google Scholar]
- Waldschmidt, M. Open Diophantine problems. Mosc. Math. J.
**2004**, 4, 245–305. [Google Scholar] [CrossRef] [Green Version] - Brindza, B. Zeros of polynomials and exponential Diophantine equations. Comp. Math
**1987**, 61, 137–157. [Google Scholar] - Hajdu, L.; Laishram, S.; Tengely, S. Power values of sums of products of consecutive integers. Acta Arith.
**2016**, 172, 333–349. [Google Scholar] [CrossRef] [Green Version] - Subburam, S. On the Diophantine equation la
^{x}+ mb^{y}= nc^{z}. Res. Number Theory**2018**, 4, 25. [Google Scholar] [CrossRef] - Subburam, S. A note on the Diophantine equation (x + a
_{1})^{r1}+ (x + a_{2})^{r2}+ ⋯ + (x + a_{m})^{rm}= y^{n}. Afrika Mat.**2019**, 30, 957–958. [Google Scholar] [CrossRef] - Bazsó, A. On linear combinations of products of consecutive integers. Acta Math. Hung.
**2020**, 162, 690–704. [Google Scholar] [CrossRef] - Bazsó, A.; Berczes, A.; Hajdu, L.; Luca, F. Polynomial values of sums of products of consecutive integers. Monatsh. Math
**2018**, 187, 21–34. [Google Scholar] [CrossRef] [Green Version] - Tengely, S.; Ulas, M. Power values of sums of certain products of consecutive integers and related results. J. Number Theory
**2019**, 197, 341–360. [Google Scholar] [CrossRef] [Green Version] - Subburam, S. The Diophantine equation (y + q
_{1})(y + q_{2})⋯(y + q_{m}) = f(x). Acta Math. Hung.**2015**, 146, 40–46. [Google Scholar] [CrossRef] - Laurent, M.; Mignotte, M.; Nesterenko, Y. Formes linéaires en deux logarithmes et determinants d’interpolation. J. Number Theory
**1995**, 55, 285–321. [Google Scholar] [CrossRef] [Green Version] - Srikanth, R.; Subburam, S. On the Diophantine equation y
^{2}= ∏_{i≤8}(x + k_{i}). Proc. Indian Acad. Sci. (Math. Sci.)**2018**, 128, 41. [Google Scholar] [CrossRef] - Subburam, S.; Togbe, A. On the Diophantine equation y
^{n}= f(x)/g(x). Acta Math. Hung.**2019**, 157, 1–9. [Google Scholar] [CrossRef] - Szalay, L. Superelliptic equation y
^{p}= x^{kp}+ a_{kp−1}x^{kp−1}+ ⋯ + a_{0}. Bull. Greek Math. Soc.**2002**, 46, 23–33. [Google Scholar] - Laurent, M. Linear forms in two logarithms and interpolation determinants II. Acta Arith.
**2008**, 133, 325–348. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Subburam, S.; Nkenyereye, L.; Anbazhagan, N.; Amutha, S.; Kameswari, M.; Cho, W.; Joshi, G.P.
On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers. *Mathematics* **2021**, *9*, 1813.
https://doi.org/10.3390/math9151813

**AMA Style**

Subburam S, Nkenyereye L, Anbazhagan N, Amutha S, Kameswari M, Cho W, Joshi GP.
On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers. *Mathematics*. 2021; 9(15):1813.
https://doi.org/10.3390/math9151813

**Chicago/Turabian Style**

Subburam, S., Lewis Nkenyereye, N. Anbazhagan, S. Amutha, M. Kameswari, Woong Cho, and Gyanendra Prasad Joshi.
2021. "On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers" *Mathematics* 9, no. 15: 1813.
https://doi.org/10.3390/math9151813