Solution of Exponential Diophantine Equation Involving Fermat Primes

Abstract

This paper investigates the solutions of specific exponential Diophantine equations involving Fermat primes. Through a structured case-by-case analysis, integer solutions are identified for equations of the form , where are non-negative integers. Theorems and proofs are provided to demonstrate the validity of these solutions. The study confirms that certain triplets, such as (1,0,2), (2,2,5), and others, satisfy the given equations under distinct mathematical constraints. These results extend the theoretical framework of Diophantine analysis and may have implications for cryptographic applications and computational number theory.

Introduction

The paper explores exponential Diophantine equations, a subset of number theory problems where integer solutions are sought for equations involving variables in the exponents. Specifically, it investigates equations connected to Fermat primes and derives their solutions using structured case-based proofs. The goal is to identify patterns and constraints that govern these equations, contributing to the broader understanding of number theory.

Key Concepts and Theorems:

1. Catalan’s Conjecture (Mih?ilescu’s Theorem):

   • States that the only two consecutive natural number powers are 23=82^3 = 823=8 and 32=93^2 = 932=9.

   • Proved in 2002 and relates to uniqueness in solutions to exponential Diophantine equations.

Theorems and Their Solutions:

Each theorem analyzes a specific form of an exponential Diophantine equation through three case studies, and identifies valid integer solutions.

Theorem 1:

Equation:
ax+by=ca^x + b^y = cax+by=c
Solutions:

• (1, 0, 2)

• (2, 2, 5)

Theorem 2:

Equation:
ax+by=3a^x + b^y = 3ax+by=3
Solution:

• (1, 1, 3)

Theorem 3:

Equation:
ax+by=9a^x + b^y = 9ax+by=9
Solution:

• (1, 3, 9)

Theorem 4:

Equation:
ax+by=129a^x + b^y = 129ax+by=129
Solution:

• (1, 7, 129)

Methodology:

• For each equation, the solution space is divided into three cases, based on different assumptions about the values of variables a,b,x,ya, b, x, ya,b,x,y.

• Logical deductions and substitutions are used to verify or eliminate possible combinations.

• The method ensures that only non-negative integer solutions are considered, adhering to Diophantine equation constraints.

Conclusion

The study successfully determines integer solutions for a class of exponential Diophantine equations involving Fermat primes. By dividing the proofs into systematic cases, we demonstrate the constraints under which such equations hold true. The solutions obtained contribute to the ongoing research in Diophantine analysis and provide a foundation for further exploration in number theory. Future research may focus on generalizing these findings to broader classes of equations or exploring their applications in computational fields.

References

[1] G.Jeyakrishnan and G.Komahan, “On the diophantine equation ”, Acta Ciencia indica mathematics, vol 2, 195-196, 2016. [2] G.Jeyakrishnan and G.Komahan, “On the diophantine equation ”, International Journal for Innovative Research in science & Technology, vol 3, issue 9, 119-120, Feb 2017. [3] Ivan Niven, Herbert, S.Zuckerman and Hugh L.Montgomery, “An Introduction to the theoryof Numbers”, John Wiley and Sons Inc, New York 2004. [4] L.K.Hua, “Introduction to the Theory of Numbers”, Springer-Verlag, Berlin-New york, 1982. [5] P.Saranya and G.Janaki, “On the exponential diophantine equation ”, International Research Journal of Engineering and Technology, vol 4, issue 11, 1042-1044, Nov 2017 [6] R.D.Carmichael, “History of theory of numbers and diophantine analysis”, Dover Publication, New york, 1959. [7] Somchitchotchaisthit, “On the diophantine equation , where p is a prime number”, American Jr. of mathematics and sciences, vol-1, issue 1, Jan 2012. [8] Saranya, C., and G. Yashvandhini. \"Integral Solutions of an Exponential Diophantine Equation .\" International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 9, no. 4, 2022, pp. 50-52 [9] Saranya, C., Janani M., and Salini R. \"Integer Solutions of Some Exponential Diophantine Equations.\" The Mathematics Education, vol. 58, no. 1, Mar. 2024.

Copyright

Copyright © 2025 C. Saranya, A. Swetha Shree. 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.