Question

List examples of equations similar to the Pell's equation, ie. any  Diophantine equation other then Pell's.

Answer 1

a)

Linear Combination

A Diophantine equation in the form is known as a linear combination. If two relatively prime integers and are written in this form with , the equation will have an infinite number of solutions. More generally, there will always be an infinite number of solutions when . If , then there are no solutions to the equation. To see why, consider the equation . is a divisor of the LHS (also notice that must always be an integer). However, will never be a multiple of , hence, no solutions exist.

Now consider the case where . Thus, . If and are relatively prime, then all solutions are obviously in the form for all integers . If they are not, we simply divide them by their greatest common divisor.

Pythagorean Triples

Main article: Pythagorean triple

A Pythagorean triple is a set of three integers that satisfy the Pythagorean Theorem, . There are three main methods of finding Pythagorean triples:

Method of Pythagoras

If is an odd number, then is a Pythagorean triple.

Method of Plato

If , is a Pythagorean triple.

Babylonian Method

For any , is a Pythagorean triple.

Sum of Fourth Powers

A equation of form has no integer solutions, as follows: We assume that the equation does have integer solutions, and consider the solution which minimizes . Let this solution be . If then their GCD must satsify . The solution would then be a solution less than , which contradicts our assumption. Thus, this equation has no integer solutions.

If , we then proceed with casework, in .

Note that every square, and therefore every fourth power, is either or . The proof of this is fairly simple, and you can show it yourself.

Case 1:

This would imply , a contradiction.

Case 2:

This would imply , a contradiction since we assumed .

Case 3: , and

We also know that squares are either or . Thus, all fourth powers are either or .

By similar approach, we show that:

, so .

This is a contradiction, as implies is odd, and implies is even. QED