integer equation
Source: Korean 1994
March 11, 2008
number theory unsolvednumber theory
Problem Statement
Consider the equation y^2\minus{}k\equal{}x^3, where is an integer.
Prove that the equation cannot have five integer solutions of the form
(x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4).
Also show that if it has the first four of these pairs as solutions, then 63|k\minus{}17.