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The exponential Diophantine equation 2 x + b y = c z
1Luoyang Institute of Science and Technology, Luoyang, Henan 471023, China.
Thescientificworldjournal
|June 25, 2014
Summary
This study classifies integer solutions for the equation 2^(x) + b^(y) = c^(z). It proves that for c = b + 2, the only solution is (1,1,1), with specific exceptions.
Area of Science:
- Number Theory
- Diophantine Equations
Background:
- The study addresses the Diophantine equation 2^(x) + b^(y) = c^(z).
- Focuses on coprime odd positive integers b and c, where min{b, c} > 1.
Purpose of the Study:
- To provide a complete classification of all positive integer solutions (x, y, z) for the given equation.
- To analyze the specific case where c = b + 2.
Main Methods:
- Employs an elementary approach to analyze the equation.
- Involves classifying integer solutions based on the properties of b and c.
Main Results:
- A full classification of solutions for 2^(x) + b^(y) = c^(z) is presented.
- For c = b + 2, the equation has a unique solution (1,1,1), barring two specific exceptions: (b, x, y, z) = (89,13,1, 2) and (2^r - 1, r + 2, 2, 2) for r ≥ 2.
Conclusions:
- The research provides a comprehensive understanding of the solutions to the analyzed Diophantine equation.
- Highlights the specific conditions under which unique or exceptional solutions arise.
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