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For a rational prime $p>3$, denote by $\zeta$ a fixed primitive root of unity of degree $p$, and let ${\mathbb K}={\mathbb Q}(\zeta)$ be the $p$th cyclotomic field. Consider the set of all non-zero elements of $\mathbb K$ of the form $a_0+a_1\zeta+\dotsb+a_{p-2}\zeta^{p-2}$ with $a_0,\dotsc,a_{p-2}\in\{0,1\}$. There are finitely many such elements; hence, the set of all their (ideal) prime divisors is finite, too. Can anything intelligent be said about this set? Say, what are the rational primes lying below these ideal primes?

Thanks in advance for any insights!

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    $\begingroup$ I would guess that the resulting norms look reasonably random for most choices of binary coefficients, aside from the obvious upper bound for the norm (and the expected value of the norm, which would be more-or-less the square root of the trivial upper bound due to random walk considerations). Have you done some experiments? $\endgroup$ Jan 20, 2015 at 18:13
  • $\begingroup$ @Joe: I haven't (except for checking $p=3$ and $p=5$). $\endgroup$
    – Seva
    Jan 20, 2015 at 19:16

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