Help...Is there an algebraic approach?

Q) Let a,ba, b and nn be integers and ab=n2+n+1ab = n^2+n+1. Prove that (ab)24n (a-b)^2 \geq 4n .

Please give a complete solution. Thank you very much

#NumberTheory

Note by Syed Hamza Khalid
11 months, 1 week ago

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1 vote

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Comments

@Zakir Husain

Yajat Shamji - 11 months, 1 week ago

I have no idea, @Syed Hamza Khalid.

@Zakir Husain, can you help him out?

Yajat Shamji - 11 months ago

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I'm trying right now!!!

Zakir Husain - 11 months ago

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

Mahdi Raza - 11 months ago

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Doesn't lead to anything sadly

Syed Hamza Khalid - 11 months ago

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Yes it doesn't

Mahdi Raza - 11 months 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).

Vinayak Srivastava - 11 months ago

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

Vinayak Srivastava - 11 months ago

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

Yajat Shamji - 11 months ago

Meaning 00 or 11.

Yajat Shamji - 11 months ago

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Not really though.

Such a condition is satisfied by the integers:

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

Syed Hamza Khalid - 11 months ago

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Well - this is out of my league...

Yajat Shamji - 11 months ago

Syed Hamza Khalid - 11 months ago

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This is a solution using the Euclidean algorithm

Syed Hamza Khalid - 11 months ago

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How you can say that nan\geq a

Zakir Husain - 11 months ago

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