perplexus.info :: Just Math : Exploring Coprime Roots in Quadratic Equations

Exploring Coprime Roots in Quadratic Equations (Posted on 2024-04-09)

For how many natural numbers n<45² are there coprime natural numbers a and b such that 45-√n is the root of the equation x²-ax+b=0?

No Solution Yet Submitted by Danish Ahmed Khan

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re: a solution if double root

| Comment 5 of 7 |

(In reply to a solution if double root by Larry)

You don't need the spreadsheet to test all 45 of those cases. If we have a double root then the quadratic is of the form f(x)=x^2-2rx+r^2.

Then apply the coprime condition, then we must have 2r is coprime to r^2. But clearly r divides both 2r and r^2.

So r can only be 1, 0 or -1. We can reduce this further from the requirement that a and b are natural numbers (positive integers). Then a=2 from r=1 is the only option.

Posted by Brian Smith on 2024-04-10 12:35:14

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