Differential geometry with the Explore plot 
vectorfield_on_sphere_in_exploreplot_with_differential_geometry.mw

@Kitonum 

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

Using Lie algebra ?
bol_en_vectorveld_mprimes_9-11-2025.mw

Try doing something with differential geometry in Maple and with other vector fields on the sphere.

An explore plot shows that it is not yet  complete  possible to work with the differential geometry code ?.
Try doing something via a maplet.

@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.

@Alfred_F 
Ai generates a procedure, but it is not straight forwards as it seems , you can decipher it ?
procedure_mprimes_number_theory_25-102025.mw

@Alfred_F 
How can you make it general , this type of calculations ?

@sand15 

You also have Maplet Builder  for working in a worksheet , which should be easy to use, but because it is so user-unfriendly, it is...
- you have to memorise all the controls right away, because there is no feedback on the mapletbuilder self about their name/meaning.

- the controls in the mapletbuilder are all very small to see.
You would think it would be a piece of cake to create your GUI with the mapletbuilder, wouldn't you?

- is it possible to have a undo  ?

The teaching material is so poorly designed that even Example 1a: Simple Maplet does not mention the starting point for the Maplet Builder. 
Another clumsy feature is that you can set up your canvas via the file menu, which is called “new” in Maple.
If there's one thing I dislike, it's trial and error as a learning strategy, because it's the most unfavourable learning strategy.
Using the Maplet Builder becomes a puzzle, and that's exactly what it shouldn't be.