## Is this pleasing polynomial irreducible?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Let:

$f(x)=x^n+2x^{n-1}+3x^{n-2}+4x^{n-3}+\ldots + (n-1)x^2+nx+(n+1)$.

Is $f(x)$ irreducible?

In light of the answers to this question, I now know that this is true when $n+1$ is prime. What about when $n+1$ is composite? I have checked a lot of cases and it seems to be true.

(Unnecessary background information: this is linked to this recent question of mine. At the point where I say:

"$g_n(x,−n)=x^2h(x)$,"

the $h(x)$ in question has the property that substituting $x-1$ for $x$ puts it in the form of $f(x)$ above. If I can show that $h(x)$ is irreducible, then I will have shown that the galois group of the original polynomial is doubly transitive. Not that I am trying to draw attention back to my original question!)

-

7
Not an answer, but: it was observed a few years ago that for $n=6$ this polynomial $x^6 + 2x^5 + 3x^4 + 4x^3 + 5x^2 + 6x + 7$, though irreducible, has non-generic Galois group: it's the transitive copy of $S_5$ in $S_6$! Anybody remember who first noted this? – Noam D. Elkies Sep 9 2011 at 18:35

## 1 Answer

It is a conjecture that for any $k < n$ one has $$\frac{d^k}{dx^k}(1+x+\cdots+x^n)$$ is irreducible in $\mathbb Z[x]$. Your question is the case $k=1$. It is known that the set of integers $n\in [0,t]$ for which the polynomial $nx^{n-1}+(n-1)x^{n-2}+\cdots+1$ is reducible has size at most $O(t^{1/3+\epsilon})$, see the paper

A. Borisov, M. Filaseta, T. Y. Lam, O. Trifonov, "Classes of polynomials having only one non-cyclotomic irreducible factor", Acta Arith. 90 (1999) 121-153

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 20, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9408736824989319, "perplexity_flag": "head"}