Dario Alpern's Alpertron is convenient for solving Pell and Pell-like equations. It can even solve the one at the heart of Archimedes' cattle problem,
$$p^2-(4)(609)(7766)(4657^2)q^2=1$$
and give its 100000-digit fundamental solution in about a minute (and that's using an old computer). However, while testing the Pell-like equation for various integer $n$,
$$x^2 - 3\big(108(3n^2)^6 - 1\big)y^2 = 3n^2\tag{1}$$
the Alpertron can solve for some, but says that $n = 5$ (among others) has no solutions. But $(1)$ in fact has a parametric solution,
$$x,y = 486n^7, n$$
So why can it solve some $n$ of $(1)$, but not others? (There is a step-by-step button which may partly explain his algorithm.)
$\color{green}{Edit\, (Nov.\, 24)}$
As pointed out by Will Jagy in his answer below, the problem seems to be that $x,y$ of $(1)$ have a common factor. However, Alpertron also can't solve,
$$x^2-dy^2 = 32\tag{2}$$
for $d=761$ (co-prime $x,y = 469, 17$), $d=1489$ ($x,y = 39,1$), and many others. Thus while it is an excellent source, if it says "No solutions", let the user be aware that with its present code, it can be mistaken.
P.S. I've tried emailing Alpern about this bug, but he seems to be using an old comment/guestbook which retired April 2012.