Solutions to a Lebesgue–nagell Equation

Abstract

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

On the diophantine equation $x^2+2^a3^b73^c=y^n $

“…With the development of modern tools such as primitive divisor theorem, modular approach, or computational techniques, many authors investigated the above equation when k ≥ 1, see for example [1,2,3,7,11,13,14,15,16,17,18,19,20,22]. Especially, the cases (p 1 , p 2 , p 3 ) = (2,3,11), (2,11,19), (2,3,19), (2,3,17), (5,13,17) are considered in [6,9,10,11] and [21], respectively. For more information and the rich literature on equation (1), we refer to an excellent survey [12] and the 359 references therein.…”