Aurifeuillean factorization in Lucas sequences
 

I have noticed that the Aurifeuillean identity [TEX]L_{5k}=L_k(5F_k^2+5F_k+1)(5F_k^2-5F_k+1)[/TEX] have analogues in other Lucas sequences, for example


[TEX]P=3,Q=-1: \frac{V_{13k}}{V_k}=\left(13^3U_k^6+13^2U_k^4-4\cdot13U_k^2+1\right)^2-\left(13U_k(13U_k^2-1)^2\right)^2, k \text{ odd}[/TEX]
and
[TEX]P=1,Q=2: \frac{U_{7k}}{7U_k}=\left(2^{k-1}V_k\right)^2-\left(7U_{k-1}U_k U_{k+1}\right)^2.[/TEX]


Are there any known results on such factorization?

[url]https://en.m.wikipedia.org/wiki/Lucas_sequence#Examples[/url] may give you a way to figure that out.

Something even more strange: certain sequences possess a "double Aurifeullian factorization". For instance: if [TEX]P=2[/TEX] and [TEX]Q=10[/TEX], Then for odd [TEX]k[/TEX] we have


[TEX]V_{10k}=V_{2k}(A+B+C+D)(A+B-C-D)(A-B+C-D)(A-B-C+D),[/TEX]


where


[TEX]
A=6U_{k}V_{k}, B=5\cdot10^{(k-1)/2}V_{k}, C=3\cdot 10^{(k+1)/2}U_{k}, D=3\cdot 10^{k}.

[/TEX]