As such we know max rational points on a line can be atmost one in which slope is irrational . I wonder if there are examples of line equation which has no rational point? Is it possible like taking √2x +√3y = 5 ? I dont have a proof yet .
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2Your example is fine. Suppose $(x_0,y_0)$ was a rational point on that line. Now square both sides and deduce that $\sqrt 6\in \mathbb Q$, a contradiction. – lulu Apr 07 '22 at 20:32
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Thanks @lulu understood – Orion_Pax Apr 07 '22 at 20:34
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Note that you ought to handle the cases where either $x_0$ or $y_0$ is $0$, but those cases are easy. – lulu Apr 07 '22 at 20:40
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A simpler example would be $x + y = \sqrt 2$, If $x$ is rational, then $y$ is not. – Paul Sinclair Apr 08 '22 at 17:58
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Nice example @PaulSinclair – Orion_Pax Apr 10 '22 at 09:12
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Thanky you. I agree with lulu that your own example is fine, though. I was just pointing out a different approach. An even simpler example is $x = \sqrt 2$ ($y$ varies freely)> – Paul Sinclair Apr 10 '22 at 12:55