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This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.

Let q be an odd prime. Let c > 1 and t be positive integers such that qt + 1 = 2c². Using elementary method and a result due to Ljunggren concerning the Diophantine equation xn-1/x-1 = y², we show that the Diophantine equation x² + qm = c2n has the only positive integer solution (x, m, n) = (c² - 1, t, 2). As applications of this result some new results on the Diophantine equation x² + qm = cn and the Diophantine equation x² + (2c - 1)m = cn are obtained. In particular, we prove that Terai's conjecture is true for c = 12,24. Combining this result with Terai's results we conclude that Terai's conjecture is true for 2 ≥ c ≥ 30 [ABSTRACT FROM AUTHOR]

Let q be an odd prime. Let c > 1 and t be positive integers such that qt + 1 = 2c². Using elementary method and a result due to Ljunggren concerning the Diophantine equation xn-1/x-1 = y², we show that the Diophantine equation x² + qm = c2n has the only positive integer solution (x, m, n) = (c² - 1, t, 2). As applications of this result some new results on the Diophantine equation x² + qm = cn and the Diophantine equation x² + (2c - 1)m = cn are obtained. In particular, we prove that Terai's conjecture is true for c = 12,24. Combining this result with Terai's results we conclude that Terai's conjecture is true for 2 ≥ c ≥ 30 [ABSTRACT FROM AUTHOR]

Let a, b, c be relatively prime positive integers such that ap + bq = cr with fixed integers p, q, r ≥ 2. Terai conjectured that the equation ax + by = cz has no positive integral solutions other than (x, y, z) = (p, q, r) except for specific cases. Most known results on this conjecture concern the case where p = q = 2 and either r = 2 or odd r ≥3. In this paper, we consider the case where p = q = 2 and r > 2 is even, and partially verify Terai's conjecture. [ABSTRACT FROM AUTHOR]

Let a, b, c be relatively prime positive integers such that a + b = c for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a + b = c in positive integers has only the solution ( x, y, z) = ( p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases. [ABSTRACT FROM AUTHOR]

By using the lower bound of linear forms in two logarithms of Laurent (Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation (n + 1) x + (n + 2) y = n z has no positive integer solutions except for (n , x , y , z) = (3 , 1 , 1 , 2) . This proof is very different from Le (J. Yulin Teachers College28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation n x + (n + 2) y = (n + 1) z by using m-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158). [ABSTRACT FROM AUTHOR]