@salim-barzani 

@salim-barzani 
Initially, your question was also about the origin of the parameters, and you have now received an answer to that.
I believe you have now modified your original question and are now responding with this comment, which is not acceptable.

elleptic_curves_uitbreidingen_DEF_18-11-2025.mw
OVERVIEW OF USED OPTIONS:
• explanation=true/false     : Show/hide explanation
• steps=true/false          : Show/hide intermediate steps
• showplot=true/false       : Show/hide plot
• symbolic=true/false       : Symbolic/numerical result
• custom_label=text         : Custom label in plot
• advanced_analysis=true    : Advanced curve analysis
• compute_j_invariant=true  : Calculate j-invariant
• find_torsion_points=true  : Find torsion points
• compute_genus=true        : Calculate genus

elleptic_curves_uitbreidingen_DEFB_18-11-2025.mw

elliptic_curves_-secant_meth_proc_def_17-11-2025.mw

Let's see what this code can do ?

@Kitonum 

Thanks, this radial vector field on the sphere looks great.

@Alfred_F 

I do it now for c = -17
result4 := {{x = -710258662, y = -548507680}, {x = -1632, y = -1260}, {x = -16, y = -12}}

kwadratisch__solver_met_plots-_bereik-complex_ver_2_4-11-2025.mw
 

@Alfred_F 
Could it be right ?


 

The following ellipse equation appears to have no integer solution

(the procedure therefore does not show a plot, which is undesirable).

 

COMPREHENSIVE TEST SUITE : GeneralQuadraticSolver( )

kwadratisch__solver_met_plots_2-11-2025.mw

@dharr 
Could this be :  known_solution_151 := [1728148040, 140634693] ?

Second solution : x = 5972991296311683199, y = 486075138127903440
Controle: 5972991296311683199² - 151 × 486075138127903440² = 1

Third solution  
: x = 20644426403316189097177411880, y = 1680019594496931198149880507
Controle: 20644426403316189097177411880² - 151 × 1680019594496931198149880507² = 1

In short: The solutions to Pell's equation correspond one-to-one with the units in the number field ℚ(√n), and the fundamental solution gives precisely the fundamental unit.

The growth appears almost random - small values of n can produce enormous solutions, while some larger n values surprisingly yield relatively small solutions. This unpredictability makes Pell's equation particularly challenging from both computational and theoretical perspectives.

Some solutions for the ellipse
diophantische_vergelijkingen_-mprimes_2-11-2025.mw

travelling_wave_met_karateristieken_exploreplot_via_mprimes_vraag_29-10-2025.mw

I would also like to do it via a maplet, but as mentioned earlier, this is complicated and the maplet builder is not user-friendly.

@Alfred_F
Ask copilot...
Can you provide this proof in a more mathematical way, as a mathematician would do for this question?
Or give a general proof for this question?

@Alfred_F 

AI can create surprising combinations, offer new perspectives, and help people move beyond their usual ways of thinking. It can be a catalyst for human creativity, not a replacement.

Deepseek is a good ai and free !, but limited  in use, but last weeks it is responding always.
I got the impression that Deepseek is more clever than a payd subscription i use from ChatGpt
By the way there is also a manual solution for your puzzle task , by ai  to get. 

@Alfred_F 
Using code from @ nm 
procedure_2_nuber_theory_code_gebruikt_25-1-2025.mw

AI can help you increase your knowledge of procedures in the approach and use of Maple code.

It is essentially a sparring partner.

You can focus on concepts, while the AI takes care of the detailled coding.

Maplesoft, the developer of Maple, is investing heavily in the application of AI.