Diophantine linear equation system

[Gunther]

Here's a little puzzle that fits well between the years 2023 and 2024. A few years ago, this was a task
at the International Mathematical Olympiad. But don't worry; the problem is really not difficult to solve.
I've modified it slightly. Here's the exercise. The following system of linear equations is to be solved:

• x  + yz = 2024
• xy + z  = 2023

x, y, z should be natural numbers and >= 1.

I posted it here in the Soap Box because there is no place for it in other subforums. I wish the
interested members a lot of fun with the solution.

[raymond]

    
Interesting "little" algebra problem. Reminds me a bit of high school. 

[Gunther]

Raymond,

Interesting "little" algebra problem. Reminds me a bit of high school. 

Yes, indeed. Unfortunately we have 3 variables and only 2 equations; but there is additional information
about x, y, z. The standard methods (LU decomposition, Wilkinson orthogonalization, etc.) wo'nt work
here. You have to be creative. But once the right idea has been found, the solution is very simple.

I don't think it will be very long before the first correct solution is posted here. So let's wait a
little while longer.

[jack]

"Extended Euclidean Algorithm" ?

[Gunther]

Jack,

thank you for your answer.

"Extended Euclidean Algorithm" ?

You can try that. But I don't think it will work. Incidentally, there is really only one solution.

[HSE]

Hi Gunther,

I failed to cheat 

Genetic Algorithm is trapped in a local solution:

Code Select Expand

x 22,2193052890
y 90,0463773843 ec1 = 2023,998484
z 22,2305353835 ec2 = 2022,998485

interaccions = 12000000000;

But very nice to play with this 10 years old programs
(specially because I will need to use them very soon  )

Regards, HSE

[Gunther]

HSE,

nice try, but wrong result.

As I have already written: You have to be creative.

[jack]

I solved it by cheating 

[Gunther]

Jack,

I solved it by cheating 

why not? What's your cheating result?

[jack]

x=674, y=2, z=675 my cheat https://youtu.be/fd12cV8ZFvA?si=jkIJjj-IvJvrBtyH

[Gunther]

Okay. Check your PM.

[raymond]

My son was over for dinner with us yesterday. Lately, he has been playing with AI to create web pages for his employer. We couldn't resist feeding it this problem which only requires basic high school algebra and a half-ounce of analytical logic.

As I expected, it did have a basic knowledge of algebra but was totally lacking the required logic. After subtracting one equation from the other, it simply tried  a "brute force" approach starting with x=1 and then iterating the y and z values to fit the equations.

Obviously, it NEVER got to a solution. So much for AI.    

[HSE]

Hi Raymond,

Obviously, it NEVER got to a solution. So much for AI.

It's a good trick  

The shame is AI don't know assembly 

Code Select Expand

diophantine proc
    local x_val:qword, y_val:qword, z_val:qword

    mov r8,  2024
    @@x:
        mov r9,  2024
        @@y:
            mov r10, 2024
            @@z: 

                mov rax, r9
                mov rcx, r10
                mul rcx
                add rax, r8
                cmp rax, 2024
                jne noes

                mov rax, r8
                mov rcx, r9
                mul rcx
                add rax, r10
                cmp rax, 2023
                je solution

          noes:

            dec r10
            jnz @@z
        dec r9
        jnz @@y
    dec r8
    jnz @@x
    conout "This fail!",lf,lf
    ret

  solution:
    mov x_val, r8   
    mov y_val, r9   
    mov z_val, r10
   
    conout " x = ", str$(x_val),lf
    conout " y = ", str$(y_val),lf
    conout " z = ", str$(z_val),lf,lf

    ret
diophantine endp

But process take almost 3 seconds here 

[NoCforMe]

Obviously, it NEVER got to a solution. So much for AI.

It's a good trick  
The shame is AI don't know assembly 

I know you were kinda sorta joking, but really, what possible difference could that make, whether AI "knows" assembly? The language used to code a solution makes no difference; they could use COBOL for all that matters. The point is it hasn't a clue about how to go about solving this problem.

Like the man said, so much for AI. (Artificial ignorance in my book.) Given all the real problems needing solutions in the world, a complete waste of time.

BTW, looking at your code, I have no clue how that works. Do you mind explaining it?

[HSE]

Hi NoC,

BTW, looking at your code, I have no clue how that works. Do you mind explaining it?

Code have 3 loops exploring values between 2024 and 1 for each variable. In inner loop results of equations are evaluated. Just that.

No macros, you can see