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Is it true that $[n]_q! + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$ ?

I checked that this is true for $n$ up to $20$.

Here $[n]_q! := 1 (1 + q) (1 + q + q^2) \cdots (1 + q + \cdots + q^{n-1})$ is the q-factorial.

Penchez
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  • Still true upto $n=60$ – Peter Feb 23 '19 at 17:18
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    Verified for $n\leq 117$. Also tried most common irreducibility criteria and only ones that worked are not scalable to generic $n$ (they need specific and often large prime for each $n$ to prove). Then, monic polynomial with prime constant coefficient (this one is always with $p=2$) can be reducible only if it has roots inside (or on) as well as outside of unit circle (interesting observation by @QiaochuYuan ). Unfortunately these polynomials satisfy that condition so you cannot rule out reducibility this way either... – Sil Feb 24 '19 at 08:28
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    Now posted to MO, https://mathoverflow.net/questions/324865/irreducibility-of-q-factorial-plus-1 – Gerry Myerson Mar 07 '19 at 22:19

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