The Search for Non-Negative Integer Solutions to Exponential Diophantine Equation

Abstract

In this manuscript, an exponential Diophantine equation ?(3?^2+5)?^x+?(6?^2+11)?^y=?^2,for some selected choices of non-negative integers and ? ,??Z is scrutinized for all the amalgamation ofx +y = 1,2,3 and proved that the essentialsolutions are (x,y,?,?)={(1,0,1,3),(1,0,? ?_n+3? ?_n,3? ?_n+3 ? ?_n ),(0,1,2,6),(0,1,2(?_n ) ?+6(?_n ) ?,6(?_n ) ?+12(?_n ) ? ),(1,1,1,5),(2,1,t,3?^2+6)}where? ?_n ,? ?_n ,(?_n ) ?,(?_n ) ?,n?0 aregeneral solutions of some peculiar Pell equations?^2 = D?^2 +6 where D=3,6 and t?Z.

Introduction

The study explores the Diophantine equation:

(3λ2+5)x+(6λ2+11)y=μ2(3\lambda^2 + 5)^x + (6\lambda^2 + 11)^y = \mu^2(3λ2+5)x+(6λ2+11)y=μ2

with constraints:

• x,y∈Z≥0x, y \in \mathbb{Z}_{\geq 0}x,y∈Z≥0? (non-negative integers)

• λ,μ∈Z\lambda, \mu \in \mathbb{Z}λ,μ∈Z

• x+y=1,2,or 3x + y = 1, 2, \text{or } 3x+y=1,2,or 3

It aims to find all integer solutions for these small values of x+yx + yx+y, using techniques such as:

• Pell equations

• Recurrence relations

• Case analysis

• Algebraic factorization

???? Main Results (Theorem)

The study proves that the given Diophantine equation has many integer solutions when x+y=1,2,3x + y = 1, 2, 3x+y=1,2,3, including general families of solutions generated from Pell equations. The key results are:

? Known Solutions (For small x+yx + yx+y):

• (x,y,λ,μ)=(1,0,1,3)(x, y, \lambda, \mu) = (1, 0, 1, 3)(x,y,λ,μ)=(1,0,1,3)

• (1, 0, \mu?_n + 3\lambda?_n, 3\mu?_n + 3\lambda?_n)

• (0,1,2,6)(0, 1, 2, 6)(0,1,2,6)

• (0,1,2μ^n+6λ^n,6μ^n+12λ^n)(0, 1, 2\hat{\mu}_n + 6\hat{\lambda}_n, 6\hat{\mu}_n + 12\hat{\lambda}_n)(0,1,2μ^?n?+6λ^n?,6μ^?n?+12λ^n?)

• (1,1,1,5)(1, 1, 1, 5)(1,1,1,5)

• (2,1,t,3t2+6)(2, 1, t, 3t^2 + 6)(2,1,t,3t2+6), for any t∈Zt \in \mathbb{Z}t∈Z

These solutions are derived using generalized Pell-type equations.

???? Approach and Methodology

???? Case-by-case Analysis by x+yx + yx+y:

Case 1: x+y=1x + y = 1x+y=1

• Subcase (1,0):

  • Reduces to μ2=3λ2+6\mu^2 = 3\lambda^2 + 6μ2=3λ2+6

  • General solutions derived from Pell equation μ2=3λ2+1\mu^2 = 3\lambda^2 + 1μ2=3λ2+1

• Subcase (0,1):

  • Reduces to μ2=6λ2+12\mu^2 = 6\lambda^2 + 12μ2=6λ2+12

  • Solutions from Pell equation μ2=6λ2+1\mu^2 = 6\lambda^2 + 1μ2=6λ2+1

???? Each case produces recursive sequences of solutions using fundamental Pell equations.

Case 2: x+y=2x + y = 2x+y=2

• (2,0): No integer solution (due to square + 1 ≠ square).

• (1,1): Single solution found: (1,1,1,5)(1,1,1,5)(1,1,1,5)

• (0,2): No integer solution.

Case 3: x+y=3x + y = 3x+y=3

• (3,0): No integer solution.

• (2,1): Infinite solutions of the form (2,1,t,3t2+6)(2,1,t,3t^2 + 6)(2,1,t,3t2+6)

• (1,2) and (0,3): No integer solutions (confirmed through analysis and factorization techniques).

???? Use of Pell Equations

The core technique involves solving:

• μ2=3λ2+1\mu^2 = 3\lambda^2 + 1μ2=3λ2+1

• μ2=6λ2+1\mu^2 = 6\lambda^2 + 1μ2=6λ2+1

These yield general solutions via:

\mu_n = \frac{(a + \sqrt{D})^{n+1} + (a - \sqrt{D})^{n+1}}{2}
]

\lambda_n = \frac{(a + \sqrt{D})^{n+1} - (a - \sqrt{D})^{n+1}}{2\sqrt{D}}
]

Where D=3D = 3D=3 or 666, depending on the case.

These values generate families of solutions for (x,y,λ,μ)(x,y,\lambda,\mu)(x,y,λ,μ), summarized in tables with computed values for small nnn.

Conclusion

The study of findings in this paper demonstrates that there is aninfinite number of solutions to the exponential Diophantineequation(?3??^2+5)^x+ (?6??^2+11)^y= ?^2where ? ,??Z , for specific combinations of x + y = 1,2,3.In this manner, one can search for solutions to the equations behavior for x + y > 3.

References

Dickson, E. Leonard, “History of the Theory of Numbers.”, Vol. 2,Chelsea Publishing Company, 1952. Mordell, J. Louis,” Diophantine Equations. “, Academic Press, 1969. Nobuhiro Terai,“On the Exponential Diophantine Equation(?4m?^2+1)^x+(?5m?^2-1)^y=z^2.”International Journal of Algebra, Vol.6,No.23,2012,1135 - 1146. A. Suvarnamani,\"On the Diophantine Equation p^x+ (p+1)^y= z^2. \" International Journal of Pure and Applied Mathematics, Vol. 94,No. 5,2014,689-692. Bushtein. “All the Solutions of the Diophantine Equationp^x+ (p+4)^y= z^2where p,(p+4) are Primes and x + y = 2,3,4.”Annals of Pure and Applied Mathematics, Vol. 16, No. 1,2018,241-244. SaniGupta, Satish Kumar, and Hari Kishan. “On the Non-Linear Diophantine Equation p^x+ (p+6)^y= z^2. “Annals of Pure and Applied Mathematics, Vol. 18, No. 1,2018,125-128. Juanli Su and Xiaoxue Li, “The Exponential Diophantine Equation(?4m?^2+1)^x+(?5m?^2-1)^y=z^2.” Hindawi Publishing Corporation, Vol.2014, Article ID 670175,5 pages. S. Saranya and V. Pandichelvi, “Frustrating solutions for two exponential Diophantine equations p^a+?(p+3)?^b-1=c^2and (?p+1)?^a-p^b+1=c^2.”Journal of Xi’an Shiyou University, Natural Science Edition, Vol.17,No.05,2021,147-156. W. S. Gayo and J. B. Bacani. “On the Diophantine equation M_x^p+(M_q+1)^q=z^2.” European Journal of Pure and Applied Mathematics,Vol. 14,No.2,2021,396-403. R. Dokchan and A. Pakapongpun. “On the Diophantine p^x+ (p+20)^y= z^2.”International Journal of Mathematics and Computer Science, Vol.6,No.1,2021,179-183. WachirarakOrosram, KitsanuphongMakonwattana, SaichonKhongsawat. “On the Diophantine Equation (p+4n)^x+ (p)^y= z^2.” European JournalofPure and Applied Mathematics, Vol.15, No 4,2022,1593-1596. SutonTadee and Apirat Siraworakun.” Non-existence of positive integer solutions of the Diophantine equation p^x+ (p+2q)^y= z^2 where p,q and p+ 2q are prime numbers.”European Journal of Pure and Applied Mathematics, Vol.16,No.2,724-735,2023. ChokchaiViriyapong and NonglukViriyapong. On the diophantine equationa^x+ (a+2)^y= z^2 , where a ? 5 (mod 21). International Journal of Mathematics &Computer Science, Vol. 18, No.3,2023.

Copyright

Copyright © 2025 B. Umamaheswari, V. Pandichelvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.