The Scientific World JOURNAL· 2015Q2
The Diophantine Equation 8<sup><i>x</i></sup> + <i>p</i><sup><i>y</i></sup> = <i>z</i><sup>2</sup>
- 672citations
- Q2SCImago
- 2015year
Short summary
This paper proves that for a fixed odd prime p, the Diophantine equation 8^x + p^y = z^2 has no positive integer solutions when p is congruent to ±3 (mod 8).
AI-generated from the title and abstract; the full text is not read.
Key points
- No positive integer solutions exist for 8^x + p^y = z^2 when p ≡ ±3 (mod 8).
- For p ≡ 7 (mod 8), solutions are linked to odd primes q where q ≡ 1 (mod 3).
- For p ≡ 1 (mod 8) and p ≠ 17, there are at most two positive integer solutions.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3(mod 8), then the equation 8 (x) + p (y) = z (2) has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod 8), then the equation has only the solutions (p, x, y, z) = (2 (q) - 1, (1/3)(q + 2), 2, 2 (q) + 1), where q is an odd prime with q ≡ 1(mod 3); (iii) if p ≡ 1(mod 8) and p ≠ 17, then the equation has at most two positive integer solutions (x, y, z).
The authors' abstract, as published at the source. The Scientific World JOURNAL, 2015 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics