What is vieta jumping? Please explain with proof.
What is vieta jumping? Please explain with proof.
V ieta jumping is a nickname for a particular kind of descent method that has become quite popular in higher level math Olympiad number theory problems. Like other instances of descent, it occurs when you have to solve a Diophantine equation (or system of equations, congruences or inequalities) whose solutions have some recursive structure. It is important to understand that Vieta jumping is not a technique that you can choose to use or not to use: most Vieta jumping problems can only be solved by Vieta jumping (else would require a tremendous amount of work).
The method of Vieta jumping, also known as root flipping, can be very useful in problems involving divisibility of positive integers. The idea is to assume the existence of a solution for which the statement in question is wrong, and then to consider the given relation as a quadratic equation in one of the variables. Using Vieta’s formula, we can display a second solution to this equation. The next step is to show that the new solution is valid and smaller than the previous one. Then by the argument of infinite descent or by assuming the minimality of the first solution, we get a contradiction.
ieta jumping is a nickname for a particular kind of descent method that has become quite popular in higher level math Olympiad number theory problems. Like other instances of descent, it occurs when you have to solve a Diophantine equation (or system of equations, congruences or inequalities) whose solutions have some recursive structure. It is important to understand that Vieta jumping is not a technique that you can choose to use or not to use: most Vieta jumping problems can only be solved by Vieta jumping (else would require a tremendous amount of work).
The method of Vieta jumping, also known as root flipping, can be very useful in problems involving divisibility of positive integers. The idea is to assume the existence of a solution for which the statement in question is wrong, and then to consider the given relation as a quadratic equation in one of the variables. Using Vieta’s formula, we can display a second solution to this equation. The next step is to show that the new solution is valid and smaller than the previous one. Then by the argument of infinite descent or by assuming the minimality of the first solution, we get a contradiction.
