A caboodle of Pell's equation in one?Find all the pairs of positive integers (x,y) satisfyingx2+y2−5xy+5=0

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A caboodle of Pell&#039;s equation in one?Find all the pairs of positive integers (x,y) satisfyingx2+y2−5xy+5=0

Toby Morris · Accepted Answer

It&#039;s (5y−2x)2−21y2=−20, which is a Pell type equation.I got that by completing the square:x2−5xy+y2=−5⇒(x−52y)2−214y2=−5⇒(5y−2x)2−21y2=−20So it&#039;s X2−21Y2=−20, with X=5y−2x and Y=y.

Cypeexorpjng · Accepted Answer

The equation is symmetric and it is easy to see solutions if, for example, we solve for y.x2+y2−5xy+5=0⇒y=5x±21x2−202|x|≥1Note 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 forspace−50000≤x≤50000.. Here is that &quot;sample&quot; 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±21y2−202so the x,y values should be interchangeable.

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

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