I was having free time, and was having no books around, so I tried to find a solution to Pell's equation by myself to pass time. But then I really got a strange result regarding the integer solutions of the equation. x^2 – 1 = ny^2 -------- (1) So, (x/y)^2 – (1/y)^2 = n Taking (1/y)^2 as k, this turns out the equation as (x^2)/k = n + k So, x^2 = k^2 + nk Thus, K^2 + nk – x^2 = 0 By the quadratic formula, the first value of k turns out to be imaginary, but inspecting the second value of k, and converting it in the form of y^2, it turns out to be, y^2 = 2/(n + √(n^2 + 4x)) and eventually, ny^2 + (y^2)( √(n^2 + 4x)) = 3 substituting value of ny^2 from (1), we get x^2 – 1 + (y^2)( √(n^2 + 4x))=2 This implies, x^2 + y^2(√(n^2+4x))=3 Now taking √(n^2 + 4x) as z , and also noticing that minimum value of both n and x is 1, the only value of z which fits the equation is 5, and for that x must be equal to 1, so the equation turns out to be, 1 + y√5 = 3 But in this case y cannot be an integer, so it eventually proves that pell’s equation has no integer solutions.

What is wrong with my proof ?

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TopNewestOkay thnx I got it. I HAVE FINALLY GOT A SOLUTION BY MYSELF! – Siddharth Kumar · 4 years, 7 months ago

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Please double check that you didn't make any careless calculation mistakes. There were several in your post, even allowing for errors carried forward. – Calvin Lin Staff · 4 years, 7 months ago

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You can start by fixing this step, and then work out the rest and be careful with your algebra:

"(x/y)^2 – (1/y)^2 = n Taking (1/y)^2 as k, this turns out the equation as (x^2)/k = n + k"

Specifically, note that if 1/y^2 = k, then (x/y)^2 = x^2 * k not x^2/k. – Eli Ross · 4 years, 7 months ago

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plzz help me – Siddharth Kumar · 4 years, 7 months ago

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– Calvin Lin Staff · 4 years, 7 months ago

Please check your equations and correct all the careless mistakes.Log in to reply

– Zi Song Yeoh · 4 years, 7 months ago

@Siddharth K. Can you type your proof in LaTex, I'm confusedLog in to reply

You made the mistake on the 4th line by writing \(\frac{x^2}{y^2}\) = \(\frac{x^2}{k}\), rather than the correct substitution, \(\frac{x^2}{y^2}\) = \(x^{2}\)k. If you interested in learning how the equation is derived and why it works then visit the wiki page I created: https://brilliant.org/wiki/quadratic-diophantine-equations-pells-equation/ Also hover your mouse over my equations to see how I formatted them - that should help :) – Curtis Clement · 2 years, 7 months ago

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