## Cyclotomic polynomial puzzles

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• Firstly, I m using Phi(n) to represent the cyclotomic polynomial whose index is n, assuming n is non-squarefree, and using p,q,r,... for primes. It s well
Message 1 of 43 , May 9, 2009
Firstly, I'm using Phi(n) to represent the cyclotomic polynomial whose index is n, assuming n is non-squarefree, and using p,q,r,... for primes.

It's well known that Phi(p) only have the coefficient 1.
It's also well known that Phi(pq) have coefficients bounded in absolute value by 1.

So here're the puzzles:

1a) What can be said about bounds on coefficients of Phi(pqr)?
b) What's the assymptotic growth?
c) For what pqr does Phi(pqr) have the largest coefficients (that you can find)?

2) As 1, but what if you restrict p,q,r to be simultaniously == 1 mod 4, or == 3 mod 4? Is there a significant difference between the two subsets? (Inspired by Gosper)

3a) Can you categorise p,q,r such that Phi(pqr) only has coefficients in {-1,0,1}?
b) Can you find a bigger pqr with that property than anyone else?

Enjoy!
Phil
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• ... On who regularly publishes, in mathematical journals, articles that do not derive principally from other subjects, such as physics. David (pleading not
Message 43 of 43 , May 23, 2013