Vieta jumping is a method for studying Diophantine equations by replacing one integer polynomial
root of a quadratic equation by its other
polynomial root, using Vieta's
formulas. It is often combined with infinite
descent (Medrano Martín del Campo 2021, Numberphile 2025). For fixed 
, suppose the equation
has the form
 |
where 
and 
are integers.
If 
is an integer polynomial root, then 
is another integer polynomial root, and 
. Positivity and a strict decrease of the chosen size
must be established separately when using this replacement for infinite
descent.
For example, fix an integer 
and consider
 |
For a solution in positive integers with 
, the other polynomial
root is 
.
The quadratic polynomial evaluated at 
is 
, so 
. Replacing 
by 
therefore decreases the larger coordinate
until the boundary solution 
is reached. When 
, the last positive solution
before this boundary is 
.
Reversing these steps gives all solutions in nonnegative integers, up to interchange of the coordinates.
They are 
for 
,
where 
,
, and the recurrence
equation is 
for 
.
The solutions with both coordinates positive
correspond to 
.
