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Proof Contest Day 7

Prove that there exist infinitely many positive integers \(n\) such that \(n^2+1\) has a prime divisor greater than \(2n +\sqrt{2n}\).

Note by Lakshya Sinha
8 months, 3 weeks ago

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Since Nobody posted the proof here is the proof:

Lakshya Sinha · 8 months, 3 weeks ago

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@Lakshya Sinha What is the motivation behind this? How can one go about approaching this proof? Calvin Lin Staff · 8 months, 2 weeks ago

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@Calvin Lin Well it helps in school unlike from brilliant here we have 3 chances. Lakshya Sinha · 8 months, 2 weeks ago

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@Lakshya Sinha I do not understand your comment.

I am asking how one can motivate finding the proof to this problem. An understanding of the problem should also provide some insight into why the solution has a certain approach to it, and motivate why someone should consider such a solution.

Currently, this solution is like "magic", where while the individual steps are explained, it is not clear why we chose them, or why these steps work. For example, why do we look at \( p\equiv 1 \pmod{8} \)? What is it about \( n^2 + 1 \) that tells us about prime divisors of the form \( p \equiv 1 \pmod{8} \)? Calvin Lin Staff · 8 months, 2 weeks ago

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