Is $\sum_{r=1}^nrx^{r-1}$ always irreducible (over the integers)?

Asked Oct 24 '20 at 20:06

Active Oct 31 '20 at 15:06

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Is $$\sum_{r=1}^nrx^{r-1}$$ always irreducible (over the integers)?

For example, the polynomial $1+2x+3x^2+4x^3+5x^4+6x^5+7x^6$ is of the requred form. I have no idea how to approach this. I have simply experimented with these polynomials and so far they seem to be irreducible for small values of $n$.

Thanks for your help.

asked Oct 24 '20 at 20:06

A-Level Student

6,520

1

Equivalent to whether $(nx^{n+1}-(n+1)x^n+1)/(x-1)^2$ is reducible. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Oct 24 '20 at 20:24
@TheSimpliFire sorry, I don't see why. Would you mind elaborating? – A-Level Student Oct 24 '20 at 20:25
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$\sum rx^{r-1}=d/dx(\sum x^r)$ which is a geometric series which has a closed form and then you can use the quotient rule to differentiate – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Oct 24 '20 at 20:26
@TheSimpliFire Ah, very interesting, thanks! – A-Level Student Oct 24 '20 at 20:27
If $n=p-1$ for some odd prime number $p$ then the polynomial is either a product of two factors of degree $\tfrac{n}{2}$, or it is irreducible. – Servaes Oct 24 '20 at 20:55
@Servaes do you have a proof for that? – A-Level Student Oct 24 '20 at 20:56
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@A-levelStudent I thought I did, but it turns out my proof only works when $2$ is a primitive root mod $p$. It follows from reducing mod $2$, and this question. – Servaes Oct 24 '20 at 21:00
@Servaes thanks for the partial solution. – A-Level Student Oct 24 '20 at 21:03
Student, where did you get the question? – Will Jagy Oct 24 '20 at 21:38
@WillJagy I came up with it myself, it's something I'm interested in :) – A-Level Student Oct 24 '20 at 21:39
For $n$ or $n\pm 1$ prime, play with sage or magma, check the irreducibility and if so check the ramification, trying to find some primes where it is almost Eisenstein or almost cyclotomic ? – reuns Oct 25 '20 at 00:42
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This appears to be an open problem. See the paper "Classes of polynomials having only one non-cyclotomic irreducible factor" by Borisov et al, specifically Conjecture 1 and Theorem 1. The most recent reference to this problem I could find was from 2018, where it seems that not much beyond the results of Borisov et al. are known. They show that these polynomials are asymptotically irreducible for almost all $n$. More precisely, for every $\epsilon > 0$, the number of values $n\le t$ such that $p_n$ is reducible is $O(t^{1/3+\epsilon})$. – EuYu Oct 29 '20 at 00:19
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@EuYu You should make that an answer. – mathworker21 Oct 29 '20 at 12:22
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This has been asked on the site few times, see Are derivatives of geometric progressions all irreducible?, Family of irreducible polynomials, Explain proof of irreducibility of $x^{p-1} + 2x^{p-2}+ \dots +(p-1)x + p$. (some of the posts mention reciprocal polynomial which is fine as they are irreducible if and only if this polynomial is irreducible) – Sil Oct 29 '20 at 19:22

Is $\sum_{r=1}^nrx^{r-1}$ always irreducible (over the integers)?

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