On the diophantine equation $$la^x + mb^y = nc^z$$ l a x + m b y = n

Abstract

In this paper, we give an upper bound for the solutions x, y, z of the equation in the title, of magnitude log max{a, b, c} 2+ . This yields an improvement of earlier results of Hu and Le, where the bound is cubic in log max{a, b, c}.

On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

“…In 2016, Hajdu, Laishram, and Tengely in [5] proved the above result for f (x) = x + x(x + 1) + • • • + x(x + 1) • • • (x + k). In 2018, Subburam [6] assured that, for each positive, real < 1, there exists an effectively computable constant c( ) such that max{x, y, n} ≤ c( )(log max{a, b, c}) 2+ , where (x, y, n) is a positive integral solution of the ternary exponential Diophantine equation a n = b x + c y and a, b, and c are fixed positive integers with gcd(a, b, c) = 1. In 2019, Subburam [7] provided the unconditional form of the first result for f (x) = (x + a 1 ) r 1 + (x + a 2 ) r 2 + • • • + (x + a m ) r m , where m ≥ 2; a 1 , a 2 , .…”

“…where c 2 can be bounded using the linear form of the logarithmic method in Laurent, Mignotte, and Nesterenko [12], and an immediate estimation is The result of Hajdu, Laishram, and Tengely in [5] is much stronger than the following corollary. They explicitly obtained all solutions for the values k ≤ 10 using the MAGMA computer program along with two well-known methods (See Subburam [6], Srikanth and Subburam [13], and Subburam and Togbe [14]), after proving that n ≤ 19,736 for 1 ≤ k ≤ 10. Here, we have Corollary 1.…”