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What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
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reviewed  Reject What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to? 
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revised 
What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
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comment 
What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
@Vesselin Dimitrov: Thanks for the reference. 
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awarded  Nice Question 
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revised 
What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
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What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
@Ofir: In fact, it is not true in general that $\binom pn\equiv 0\pmod{p1}$. Also, please consider fixing some typos in your proof that ${\mathcal P}_p(n)$ is a $p$th power. 
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What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
@Steve Huntsman: Looks like you have some computations done; could you post here the results? What $p$ and $n$ the values you list ($2^7,3^8,11^{10},19^9$) correspond to? (In fact, I am a bit puzzled: say, having ${\mathcal P}_p(n)=3^8$ doesn't agree with ${\mathcal P}_p(n)$ being a $p$th power, see Ofir's comment.) 
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answered  Presenting a paper: Do's and Don'ts? 
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revised 
What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to?
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asked  What are the products $\prod_{A\subset{\mathbb F}_p\colon A=n} \sum_{a\in A} \zeta^a$ equal to? 
Jan 20 
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Prime factors of $\sum_{i\in I} \zeta_p^i$
@Joe: I haven't (except for checking $p=3$ and $p=5$). 
Jan 20 
asked  Prime factors of $\sum_{i\in I} \zeta_p^i$ 
Jan 14 
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Why does this sequence converges to $\pi$?
I suppose $4$ in the righthand side of the first line of the example was supposed to be cancelled, too? 
Jan 8 
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covering high dimensional hypercube by balls
You can find a comprehensive account, say, in "Covering Codes" by G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein. 
Jan 8 
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covering high dimensional hypercube by balls
You are asking about the smallest possible size of a (not necessarily linear) binary code of covering radius $1$. There are lots of results in this direction, just google to find them. 
Jan 8 
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How large must $A$ be if $\{1, \ldots, N\} \subseteq AA$?
@Douglas Zare: It is important that $A$ needn't be contained in an interval of length $N$. 
Jan 7 
revised 
How many roots can $P(x):=\sum_s(xs)^{(p1)/2}$ have in ${\mathbb F}_p$?
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Jan 7 
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How many roots can $P(x):=\sum_s(xs)^{(p1)/2}$ have in ${\mathbb F}_p$?
@Terry: absolutely  this is a convolution, and the question is motivated by Paley graph  related busyness; but, maybe, this is the right approach to a difficult problem? Even showing that $P(x)$ cannot have $\frac{p1}2$ distinct roots, or that it cannot attain just three distinct values, would yield an (admittedly, tiny) improvement in the current bound on the clique size in the Paley graph. 
Jan 6 
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How many roots can $P(x):=\sum_s(xs)^{(p1)/2}$ have in ${\mathbb F}_p$?
@jmc: The question is about $n$ and $p$ both growing, with $n\approx\sqrt p$. 