Further results on the Diophantine equation 𝑥 2 + 16 ∙ 7 𝑏 = 𝑦 𝑛 when 𝑛 is even
This work extends the results for the Diophantine equation 𝑥 2 + 16 ∙ 7 𝑏 = 𝑦 𝑛 for 𝑛 = 2𝑟, where 𝑥, 𝑦, 𝑏, 𝑟 ∈ ℤ + . Earlier results classified the generators of solutions, which are the pair of integers (𝑥, 𝑦 𝑟 ), into several categories and presented the general formula that determines t...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Prince of Songkla University
2024-06-01
|
| Series: | Songklanakarin Journal of Science and Technology (SJST) |
| Subjects: | |
| Online Access: | https://sjst.psu.ac.th/journal/46-3/7.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | This work extends the results for the Diophantine equation 𝑥
2 + 16 ∙ 7
𝑏 = 𝑦
𝑛
for 𝑛 = 2𝑟, where 𝑥, 𝑦, 𝑏, 𝑟 ∈ ℤ
+
. Earlier
results classified the generators of solutions, which are the pair of integers (𝑥, 𝑦
𝑟
), into several categories and presented the general
formula that determines the values of 𝑥 and 𝑦
𝑟
for the respective category. The lower bound for the number of non-negative integral
solutions associated with each 𝑏 is also provided. We now extend the results and prove the necessary and sufficient conditions
required to obtain integral solutions 𝑥 and 𝑦 to the equation, by considering various scenarios based on the parity of 𝑏. We also
determine the values of 𝑛 for which integral solutions exist.
|
|---|---|
| ISSN: | 0125-3395 |