Generalizing Zsigmondy's Theorem

Main post link -> http://en.wikipedia.org/wiki/Zsigmondy's_theorem

I know that there are some generalizations of this extremely neat and beautiful result over some extensions of $\mathbb{Z}$ , more precisely I'm speaking of Gaussian Integers (proving it over any other quadratic extension would also be of great help). It would be nice if someone could point out some ideas, which I could follow to prove the result. For those who eager to help, but don't know where to start, try getting used with Cyclotomic Polynomials and classic proof of Zsigmondy's Theorem.

From the first glance the theorem may appear useless, but it really is an Olympiad gem, which sometimes cracks a quite hard question.

#NumberTheory #HelpMe! #MathCompetitions #MathProblem

Easy Math Editor

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Comments

@Nicolae Sapoval OHHH MY GODS!!!! This is a new theorem for me!!!! Thanks a lot!!!!

Aaghaz Mahajan - 2 years, 9 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}`	$\boxed{123}$