Often, one of the most helpful tools in solving Diophantine equations is referred to as "bounding." Frequently, encounters with Diophantine equations can be remarkably vague, often for a reason. This prompts the solver to make assumptions WLOG (without loss of generality) that make the problem easier to handle without fundamentally changing it. Bounding is the process of restricting values of a variable to a manageable set. The best way of illustrating this is through a few examples.
Determine all pairs of integers that satisfy the equation
Upon expansion, we reach
Now, notice that if we assume
From this, we can say that is between two consecutive perfect (integer) cubes, which is impossible. So, we can say that is nonpositive, or . Now, notice that if is a solution, then is also a solution, which implies that is nonpositive, or . Hence, the only solution is .
Solve in positive integers the equation
Dividing both sides by , we reach
Without loss of generality, suppose . This yields
Which implies . Splitting into two separate cases and applying similar logic, we can come up with some systems of equations. Upon solving, the only solutions are and all of their permutations, accounting for the symmetry of the equation.
Find all pairs of positive integers such that
We are given
Notice that . Therefore, we have to consider
Since we have
Of these equations we can see the only working value of is when and , so the only positive integer pair of solution is .