Solving quadratic or higher degree diophantine equations

I came across a problem in the set of number theory in which it involves solving of diophantine equations. (Especially this problem: https://brilliant.org/community-problem/a-number-theory-problem-by-souryajit-roy-2/?group=0RqFFfJZARDI&ref_id=211256) Some of the problems which involve diophantine equations of degree 2 or more are (really hard?) to solve. I tried to give information about the possible solutions and yes, I found some but I wasn't able to put all the solutions out. In despair, I used trial and error. And the worst case, I used Python to find all the solutions and it turned out there may be large increase from first solution to next solution. Do these kind of problems really require (not just for checking) programming language to get all the solutions? If not, what might be some techniques or theorems in which we are sure to get the solutions?

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

If you look into the solution, I mentioned that we get a Pell's equation, whose solutions grow at an exponential rate. There is no need to do a computer search, if you understand the theory of Pell's equation.

Calvin Lin Staff - 7 years, 1 month ago

Is PE suitable for all higher degree ( 2+) diophantine equations, sir?

Krishna Ar - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$