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On the Diophantine Equation a^x+(a+2)^y=z^2 |
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| รหัสดีโอไอ | |
| Creator | Suton Tadee |
| Title | On the Diophantine Equation a^x+(a+2)^y=z^2 |
| Publisher | KKU Science Journal |
| Publication Year | 2567 |
| Journal Title | KKU Science Journal |
| Journal Vol. | 52 |
| Journal No. | 1 |
| Page no. | 39 - 46 |
| Keyword | Diophantine Equation, Non-negative Integer Solution, Congruence, Mihailescu’s Theorem |
| URL Website | https://ph01.tci-thaijo.org/index.php/KKUSciJ/article/view/254845 |
| Website title | Thai Journal Online (ThaiJO) |
| ISSN | 3027-6667 |
| Abstract | In this paper, we investigated the solutions of the Diophantine equation a^x+(a+2)^y=z^2, where a is a positive integer and x,y,z are non-negative integers. Let S be the set of non-negative integer solutions (x,y,z) of the equation. The results showed that 1) if a is a prime number with a ≡ 5(mod8), then S ≡ {(0,1,sqrt{a+3})}, where sqrt{a+3} is an integer, otherwise S=Ø. 2) If a+2 is a prime number and x is even and the equation has a solution, then y = 1 and z = 2. 3) Let p be a prime number such that p ≡ 5,7(mod8) and a ≡ -2(modp). Then S = {(1,0,sqrt{a+1})}, where sqrt{a+1} is an integer, otherwise S =Ø, when it satisfies one of the following cases: case 1 a ≡ 3mod4) or case 2 there exists a prime number q such that q ≡ 3,5(mod8) and a ≡ -1(modq). |
