# Are Erdős polynomials irreducible?

Define the Erdős polynomial to be $$f_n(x):= \sum \limits_{0 \leq i,j \leq n}^{}{x^{ij}}$$ (the name is motivated by http://oeis.org/A027424).

For example for $$n=5$$, the polynomial is given by $$x^{25}+2x^{20}+x^{16}+2x^{15}+2x^{12}+2x^{10}+x^9+2x^8+2x^6+2x^5+3x^4+2x^3+2x^2+x+11$$.

Question 1: Is $$f_n(x)$$ always irreducible (over $$\mathbb{Q}$$)? If not, when is it irreducible?

It is irreducible for $$n \leq 20$$.

More generally, define for $$k \geq 2$$ the polynomial $$f_n^k(x):= \sum \limits_{0 \leq i_1,i_2,...,i_k \leq n}^{}{x^{i_1 ... i_k}}$$.

Question 2: Is $$f_n^k(x)$$ always irreducible? If not, when is it irreducible?

For $$k=3$$, it is irreducible for $$n \leq 8$$.

• If my PARI code is correct, then the first conjecture is true for $n\leq100$. Oct 6 at 21:18
• @ThomasBrowning, I have a test running in Sage which has got up to 167 so far without finding a counterexample. Oct 6 at 23:09
• It seems like all roots of $f_n(x)$ might lie outside the unit circle. Not sure how to prove it, but this is the sort of thing that can sometimes be useful for proving irreducibility. Oct 7 at 6:22
• Checked up to 232, but it's taken the past 12 hours to get from about 208 to 232 so I'm killing it now. Oct 7 at 20:17