On factoring integers of the form n=(x^D+1)(y^D+1) and n=(x^D+a_(D-2)x^(D-2)+...a_{D-2}(x^{D-2}))(y^D+b_{D-2}(y^{D-2}+...b_0)) with x,y of the same size

[Georgi Guninski]

Apologies for OT, but this might be of interest.

From our preprint [1].

> We got plausible algorithm and strong numerical evidence
that integers of the form $n=(x^D+a_{D-2}x^{D-2}+\cdots a_0)
(y^D+b_{D-2}y^{D-2}+\cdots b_0)$ with $x,y$ of the same size
and $C=\max\{a_i,b_i\}$ and $a_i \ge 0,b_i \ge 0$ can be factored in
$O(\mathrm{polynomial}(C \log(n)))$. A special case for $D>1$ is
$n=(x^D+1)(y^D+1)$ We tested thousands of testcases without failure.

1. Is this correct?
2. Is this known?
3. Can it be improved?

Example session:

Kx.<x,y>=QQ[]

B=2**400;f1=x^4+5*x^2+1;f2=y^4+14*y^2+2;x0,y0=[randint(B,2*B) for _ in
(1,2)];n=f1(x0,0)*f2(0,y0)
%time ss=guninski_fg(n,f1,f2,L=None,allsols=False,prot=1) #Wall time: 71.5 ms
print("log(sol,2)=",RR(ss[0]).log(2)) #log(sol,2)= 1602.01840756461

[1] https://www.researchgate.net/publication/395877507_On_factoring_integers_of_the_form_n_x_D_1y_D_1_and_n_x_D_a_D-2_x_D-2_a_0_y_D_b_D-2_y_D-2_b_0_with_x_y_of_the_same_size