Why can't the Alpertron solve this Pell-like equation?
Solution 1:
I have answered on MSE many times about this: using Lagrange/Gauss method, you can find the cycle of "reduced" forms in the equivalence class of $x^2 - t y^2.$ Furthermore, the disciminant of this form being $4t,$ all primitively represented values with absolute value up to $$ \frac{1}{2} \sqrt {4t} = \sqrt t $$ is found as a first coefficient of one of the forms.
Perhaps the kicker is PRIMITIVE representations. For you, $\gcd(n,486 n^7) = n. $ I would say that any solution to your problem with $n=5$ must have both $x,y$ divisible by $5.$ That would throw off the usual method. I think you should expect the same trouble for $n=5,11,17,23,29,41,\ldots$
EDIT, 5 March 2014: I just noticed the trouble about 32 with d=761, 1489. Lagrange's method easily gives these, see below where 32 occurs as the first coefficient of one of the forms (triples).
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell
Input n for Pell
761
0 form 1 54 -32 delta -1
1 form -32 10 23 delta 1
2 form 23 36 -19 delta -2
3 form -19 40 19 delta 2
4 form 19 36 -23 delta -1
5 form -23 10 32 delta 1
6 form 32 54 -1 delta -54
7 form -1 54 32 delta 1
8 form 32 10 -23 delta -1
9 form -23 36 19 delta 2
10 form 19 40 -19 delta -2
11 form -19 36 23 delta 1
12 form 23 10 -32 delta -1
13 form -32 54 1 delta 54
14 form 1 54 -32
disc 3044
Automorph, written on right of Gram matrix:
-27201 -1484800
-46400 -2532801
Pell automorph
-1280001 -35310400
-46400 -1280001
Pell unit
-1280001^2 - 761 * -46400^2 = 1
=========================================
Pell NEGATIVE
-800^2 - 761 * -29^2 = -1
=========================================
761 761
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell
Input n for Pell
1489
0 form 1 76 -45 delta -1
1 form -45 14 32 delta 1
2 form 32 50 -27 delta -2
3 form -27 58 24 delta 2
4 form 24 38 -47 delta -1
5 form -47 56 15 delta 4
6 form 15 64 -31 delta -2
7 form -31 60 19 delta 3
8 form 19 54 -40 delta -1
9 form -40 26 33 delta 1
10 form 33 40 -33 delta -1
11 form -33 26 40 delta 1
12 form 40 54 -19 delta -3
13 form -19 60 31 delta 2
14 form 31 64 -15 delta -4
15 form -15 56 47 delta 1
16 form 47 38 -24 delta -2
17 form -24 58 27 delta 2
18 form 27 50 -32 delta -1
19 form -32 14 45 delta 1
20 form 45 76 -1 delta -76
21 form -1 76 45 delta 1
22 form 45 14 -32 delta -1
23 form -32 50 27 delta 2
24 form 27 58 -24 delta -2
25 form -24 38 47 delta 1
26 form 47 56 -15 delta -4
27 form -15 64 31 delta 2
28 form 31 60 -19 delta -3
29 form -19 54 40 delta 1
30 form 40 26 -33 delta -1
31 form -33 40 33 delta 1
32 form 33 26 -40 delta -1
33 form -40 54 19 delta 3
34 form 19 60 -31 delta -2
35 form -31 64 15 delta 4
36 form 15 56 -47 delta -1
37 form -47 38 24 delta 2
38 form 24 58 -27 delta -2
39 form -27 50 32 delta 1
40 form 32 14 -45 delta -1
41 form -45 76 1 delta 76
42 form 1 76 -45
disc 5956
Automorph, written on right of Gram matrix:
-301402109537809 -23083652981172600
-512970066248280 -39287127144407089
Pell automorph
-19794264626972449 -763812428643688920
-512970066248280 -19794264626972449
Pell unit
-19794264626972449^2 - 1489 * -512970066248280^2 = 1
=========================================
Pell NEGATIVE
99484332^2 - 1489 * 2578145^2 = -1
=========================================
1489 1489
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
-
I wrote a program that displays the x,y values. Looking at the cycle above, we se that all solutions of $x^2 - 761 y^2 = 32$ are images of these under the automorphism indicated.
jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
1280001^2 - 761 46400^2 = 1
x^2 - 761 y^2 = 32
Sun May 8 08:21:00 PDT 2016
x: 469 y: 17 ratio: 27.58823529411764 SEED x: 43669 y: 1583 ratio: 27.58622867972204 SEED
Sun May 8 08:22:00 PDT 2016
x^2 - 761 y^2 = 32
Pell automorph 1280001 35310400 46400 1280001
jagy@phobeusjunior:~$