A caboodle of Pell's equation in one? Find all the pairs

omdw1u

omdw1u

Answered question

2022-03-04

A caboodle of Pell's equation in one?
Find all the pairs of positive integers (x,y) satisfying
x2+y25xy+5=0

Answer & Explanation

Toby Morris

Toby Morris

Beginner2022-03-05Added 3 answers

It's (5y2x)221y2=20, which is a Pell type equation.
I got that by completing the square:
x25xy+y2=5(x52y)2214y2=5(5y2x)221y2=20
So it's X221Y2=20, with X=5y2x and Y=y.
Cypeexorpjng

Cypeexorpjng

Beginner2022-03-06Added 6 answers

The equation is symmetric and it is easy to see solutions if, for example, we solve for y.
x2+y25xy+5=0y=5x±21x2202|x|1
Note that the absolute value of x must be at least 1 for the radical to be non-negative and therfore, for y to be real.
Given this equation, we can also see that there are 2 y-values for every valid x. There are 28 solutions for
space50000x50000.. Here is that "sample" of (x,y1,y2.)
(7369,35307,1538)(4729,22658,987)(1538,7369,321)(987,4729,206)(321,1538,67)(206,987,43)(67,321,14)(43,206,9)(14,67,3)(9,43,2)(3,14,1)(2,9,1)(1,3,2)(1,2,3)(2,1,9)(3,1,14)(9,2,43)(14,3,67)(43,9,206)(67,14,321)(206,43,987)(321,67,1538)(987,206,4729)(1538,321,7369)(4729,987,22658)(7369,1538,35307)(22658,4729,108561)(35307,7369,169166)
Note that all the negative x-values have positive counterparts and that we have a counterpart solution for x.
x=5y±21y2202
so the x,y values should be interchangeable.

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