# Help...Is there an algebraic approach?

Q) Let $a, b$ and $n$ be integers and $ab = n^2+n+1$. Prove that $(a-b)^2 \geq 4n$.

Please give a complete solution. Thank you very much

Note by Syed Hamza Khalid
4 months, 2 weeks ago

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- 4 months, 1 week ago

This is a solution using the Euclidean algorithm

- 4 months, 1 week ago

How you can say that $n\geq a$

- 4 months, 1 week ago

- 4 months, 1 week ago

Meaning $0$ or $1$.

- 4 months, 1 week ago

Not really though.

Such a condition is satisfied by the integers:

$n = 9, a = 7, b = 13 \text{ and } n = 16, a = 13, b = 21$

- 4 months, 1 week ago

Well - this is out of my league...

- 4 months, 1 week ago

All I can figure out is that $ab = 0 \backslash 1$.

- 4 months, 1 week ago

The problem seems interesting, but way more difficult than it looks! I'll try and tell if I am able to find something(very less probability though).

- 4 months, 1 week ago

Not surprisingly, I could not find a good method. Sorry!

- 4 months, 1 week ago

An approach could be to make a quadratic in $n$ and equating the discriminant greater than 0. Maybe... but I don't see whether it's truly possible

- 4 months, 1 week ago

- 4 months, 1 week ago

Yes it doesn't

- 4 months, 1 week ago

I have no idea, @Syed Hamza Khalid.

@Zakir Husain, can you help him out?

- 4 months, 1 week ago

I'm trying right now!!!

- 4 months, 1 week ago

- 4 months, 1 week ago