For the number of integer solutions of the title equation, withW≤;x (x a large parameter), an asymptotics of the form Ax log x + Bx + O(x1/2 (log x)3 (loglog x)2) is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.
Authors:Elif kizildere, Maohua le, and Gökhan Soydan
purely exponential Diophantineequation (1.1) A x + B y = C z , x , y , z ∈ ℕ for some triples ( A,B,C ) with A + B = C 2 (see [ 1,3,4,6,7,9–13 ]). See also the survey paper [ 8 ], for more details about the equation (1.1) . For example
Let a and b given unequal positive integers; it is desired to determine the positive integer solutions n and x of the equation
of the title. Some special cases have recently been considered, and here some general results and conjectures are presented.