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The Diophantine equation 8(x) + p(y) = z(2).

Lan Qi1, Xiaoxue Li2

  • 1College of Mathematics and Statistics, Yulin University, Yulin, Shaanxi 719000, China.

Thescientificworldjournal
|February 6, 2015
PubMed
Summary

This study investigates the Diophantine equation 8^x + p^y = z^2 for odd primes p. It finds conditions on p that determine the existence and number of positive integer solutions.

Area of Science:

  • Number Theory
  • Diophantine Equations
  • Algebraic Number Theory

Background:

  • Diophantine equations are polynomial equations where only integer solutions are sought.
  • Exponential Diophantine equations involve exponents that are variables.
  • Understanding the solvability of such equations is crucial in number theory.

Purpose of the Study:

  • To analyze the positive integer solutions of the exponential Diophantine equation 8^x + p^y = z^2.
  • To establish conditions on the odd prime 'p' that govern the existence and quantity of solutions.
  • To contribute to the theory of exponential Diophantine equations.

Main Methods:

  • Utilizing established results from the theory of exponential Diophantine equations.
  • Applying modular arithmetic and congruence relations to constrain possible solutions.

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  • Case-by-case analysis based on the residue of 'p' modulo 8.

    • Main Results:

    • For primes p ≡ ±3 (mod 8), no positive integer solutions exist.
    • For primes p ≡ 7 (mod 8), specific solutions are identified involving another prime q.
    • For primes p ≡ 1 (mod 8) (excluding p=17), at most two positive integer solutions are found.

    Conclusions:

    • The solvability of 8^x + p^y = z^2 is highly dependent on the properties of the prime 'p'.
    • The number of solutions is finite and can be zero, one, or two under specific conditions.
    • This research provides a comprehensive analysis of solutions for this class of exponential Diophantine equations.