The topic of this summer school was Diophantine equations, which are among the oldest studied mathematical objects. A Diophantine equation is an equation where admissible solutions are restricted to the rationals or the integers, or appropriate mathematical generalizations of such objects. The equations themselves tend to be polynomial, exponential, or a mixture of both, where variables in the ...

We discuss the feasibility of an elementary solution to the Diophantine equation of the 15 form x 2 + D = y n , where D > 1, n ≥ 3 and x > 0, called the Lebesgue–Nagell equation, which has recently been solved for 1 ≤ D ≤ 100 in [1].

In this paper, we consider the Diophantine equation b + (a+ b) + · · ·+ (a (x− 1) + b) = = d + (c+ d) + · · ·+ (c (y − 1) + d) , where a, b, c, d, k, l are given integers with gcd(a, b) = gcd(c, d) = 1, k 6= l. We prove that, under some reasonable assumptions, the above equation has only finitely many solutions.