<s>[INST] Salem numbers and Lehmer's decic Given,
$$x^{12}-x^7-x^6-x^5+1 = 0\tag1$$
This has Lehmer’s decic polynomial as a factor,
$$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1=0\tag2$$
hence one of its roots is the smallest known Salem number.  All ten roots obey the beautiful cyclotomic relation,
$$x^{630}-1=\frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{3}-1)^3(x^{2}-1)^5(x-1)^3} {(x^{35}-1)(x^{15}-1)^2(x^{14}-1)^2(x^{5}-1)^6\, x^{68}}$$
found by D. Broadhurst. But this was back in 1999 (paper here). Has anything similar for other Salem numbers been found since then?
 [/INST] Revisiting this old question, now armed with Mathematica's "Integer Relations", I find that it is quite easy to look for similar polynomial relations. For example, let $x$ be a root of Lehmer's decic, then it is also the case that,
$$x^{630}-1 = \frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{10}-1)(x^{9}-1)}{(x^{70}-1)(x^{63}-1)(x^{45}-1)(x^{42}-1)(x^{30}-1)(x^{6}-1)}$$
Given the fifth smallest known Salem number $y$ and a root of the decic,
$$y^{10}-y^6-y^5-y^4+1=0$$
then,
$$y^{210}-1 =\frac{(y^{105}-1)(y^{70}-1)(y^{42}-1)(y^{30}-1)(y^{14}-1)(y^{6}-1)(y^{3}-1)^2}{ (y^{35}-1)(y^{10}-1)(y^{7}-1)^2(y^{2}-1)\,y}$$
</s>