| Abstrakt: |
Let a , b , c , and n be positive integers such that a + b = c 2 , 2 ∤ c and n > 1 . In this paper, we prove that if gcd (c , n) = 1 and n ≥ 117.14 c , then the equation (a n 2 + 1) x + (b n 2 − 1) y = (c n) z has only the positive integer solution (x , y , z) = (1 , 1 , 2) under the assumption gcd (a n 2 + 1 , b n 2 − 1) = 1 . Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer n , if (a , b , c) = (12 , 13 , 5) , (18 , 7 , 5) , or (44 , 5 , 7) , then this equation has only the solution (x , y , z) = (1 , 1 , 2) . This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently. [ABSTRACT FROM AUTHOR] |
