What types of Diophantine equations are unsolvable?

[Guest]

On the Second Difficult Diophantine Equations with 20 Unknown Variables:


[tex](-5x_{1 } - 9 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^3x_{18}x_{20}^3

=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666[/tex]
*****

How do we solve the equation?

Hints: And we do not know the status of the second equation.

Hint: X ={1156411, -581531, -818850, -1070344, -285794, -217840, 70110, -5034, -62855, 6271, -326556, 54260, 476995, 1385342, 1619879, 118986, 989924, 33090, 743138, 554790 }

with

T = 1 762 847 010 350 434 634 953 303 025 489 436 633 560 258 864 626 134 233 475 652 600 919 910 572 466 886 276 756 453 825 447 680

[Guest]

Recall:
k = 1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,613,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666 for second equation.

[Guest]

Simple Algorithm for Mathematica:

Given X = Table[ Random[{-2580000, 2580000}], 20];
Solve T;

While[ RealAbs[T - k] > .0001T,


X = Table[ Random[{Random[{-2580000, 0}], Random[{0, 2580000}]}, 20];

Solve T;

Recall:

T(X) = [tex](3x_{1 } + 8 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 })(x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3[/tex].

[Guest]

Try: ...While[ RealAbs[T - k] > .00001T ...

Try: ... While[ RealAbs[T - k] > .000001T ...

Try: ... While[ RealAbs[T - k] > .000000001T...

etc.

[Guest]

Of course with our developed theory for Hilbert's 10th problem, we can decide the solvability of T(X) = k in polynomial time. Right?

[Guest]

In Mathematica, we use the function, RandomInteger[], blah blah blah

[Guest]

Try: ...While[ RealAbs[T - k] > .00001T ...

Try: ... While[ RealAbs[T - k] > .000001T ...

Try: ... While[ RealAbs[T - k] > .000000001T...
etc.

Update Option:

Try: ...While[ RealAbs[T - k] > .00001k ...

Try: ... While[ RealAbs[T - k] > .000001k ...

Try: ... While[ RealAbs[T - k] > .000000001k...

etc.

In any case, we need strong theory to settle our problem. Right?

[Guest]

"On the Second Difficult Diophantine Equations with 20 Unknown Variables:


[tex](-5x_{1 } - 9 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^3x_{18}x_{20}^3

=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666[/tex]
*****

How do we solve the equation?"

We hope you have fun computing/understanding our work. Good Luck!

Caveat: The attached file is quite helpful, but you must have a strong theory to solve the problem.

Reference Link:

'On the Shapes of Surfaces and the Solutions to DEs',

https://www.math10.com/forum/viewforum.php?f=63&sid=66251d3896937c19b4effcbbeb1ab9ba#:~:text=On%20the%20Shapes%20of%20Surfaces%20and%20the%20Solutions%20to%20DEs.

[Guest]

Recall: k=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666

We hope you pay close attention to the last four computations of the attached file, DE2c.pdf, of the previous post.

We have computed [tex]X_{below}[/tex] with T([tex]X_{below}[/tex]) < k.

And we have computed [tex]X_{above}[/tex] with T([tex]X_{above}[/tex]) > k...

The previous work of that pdf was to obtain those computations. We hope you can make sense of it. Good Luck!

Go Blue!

[Guest]

Update:

Recall: k=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666

We hope you pay close attention to the last few (10 or less) computations of the attached file, DE2c.pdf, of the previous post.

We have computed [tex]X_{below}[/tex] with T([tex]X_{below}[/tex]) < k.

And we have computed [tex]X_{above}[/tex] with T([tex]X_{above}[/tex]) > k...

The previous work of that pdf was to obtain those computations. We hope you can make sense of it. Good Luck!

Go Blue!

[Guest]

We believe David Hilbert's Tenth Problem has a positive answer (a polynomial-time algorithm that solves polynomial DEs).

Go Blue!

[Guest]

Update:

Recall: k=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666

We hope you pay close attention to the last few (10 or less) computations of the attached file, DE2c.pdf, of the previous post.

We have computed [tex]X_{below}[/tex] with T([tex]X_{below}[/tex]) < k.

And we have computed [tex]X_{above}[/tex] with T([tex]X_{above}[/tex]) > k...

The previous work of that pdf was to obtain those computations. We hope you can make sense of it. Good Luck!

Go Blue!

[tex]X_{below}[/tex] = {560457, 177373, -474478, 758101, 169949, 334264, -11695, 4152, -452552, -2043222, -773294, 32413, -386466, 1048283,
-1308906, 282065, -854134, 1003178, 382027, -282924} with

T([tex]X_{below}[/tex]) = 176267602100262289279516872579626741663059760431794614910869907531660240113460623290439863066291 < k.

[tex]X_{above}[/tex] = {442041, -186929, 307892, -1606456, -397597, 1603880, 541544, 246015, 481183, -2068060, 940603, 78791, 615863, 445008,
9265, 288391, 810831, -1804882, 582701, 135713} with

T([tex]X_{above}[/tex]) = 17627524068197777915295303275995726623913010126259452399858422223231325465956475214003025623756 > k.

[Guest]

Important Update:

We hope David Hilbert's Tenth Problem has a positive answer (a polynomial-time algorithm that solves polynomial DEs).

Go Blue!

[Guest]

Our holiday wish is that the math experts will share asap their insights/published results with us so that we may know the answer (positive or negative) to David Hilbert's Tenth Problem.

Or else, we will solve the problem...

"We must know. We will know!" -- David Hilbert.

[Guest]

Update:

Recall: k=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666

We hope you pay close attention to the last few (10 or less) computations of the attached file, DE2c.pdf, of the previous post.

We have computed [tex]X_{below}[/tex] with T([tex]X_{below}[/tex]) < k.

And we have computed [tex]X_{above}[/tex] with T([tex]X_{above}[/tex]) > k...

The previous work of that pdf was to obtain those computations. We hope you can make sense of it. Good Luck!

Go Blue!

[tex]X_{below}[/tex] = {560457, 177373, -474478, 758101, 169949, 334264, -11695, 4152, -452552, -2043222, -773294, 32413, -386466, 1048283,
-1308906, 282065, -854134, 1003178, 382027, -282924} with

T([tex]X_{below}[/tex]) = 176267602100262289279516872579626741663059760431794614910869907531660240113460623290439863066291 < k.

[tex]X_{above}[/tex] = {442041, -186929, 307892, -1606456, -397597, 1603880, 541544, 246015, 481183, -2068060, 940603, 78791, 615863, 445008,
9265, 288391, 810831, -1804882, 582701, 135713} with

T([tex]X_{above}[/tex]) = 17627524068197777915295303275995726623913010126259452399858422223231325465956475214003025623756 > k.

We have not exhausted our use of randomness in our computations. We may want to consider a new min and max for each component, [tex]x_{i }[/tex], of X.
For example, we select the following ranges based on our computed vectors, [tex]X_{below }[/tex] and [tex]X_{above }[/tex]:


[tex]-560457 < x_{1 } < 560457[/tex];

[tex]-186929 < x_{2 } < 186929[/tex];

[tex]-560457 < x_{3 } < 560457[/tex];

...

[tex]-282924 < x_{20 } < 2829247[/tex];


Hopefully, we can compute values, [tex]X_{below }[/tex] and [tex]X_{above }[/tex], as close as possible to the true X (if it exists). And from there, we use our developed theory (the hardest part of our problem since we have not formulated/tested a good theory based on essential ideas and methods, etc.) to decide the solvability of T(X) = k. We are making good progress.

So far, so good! Keep up the good work! Good luck!

[Guest]

Update: [tex]-282924 < x_{20 } < 282924[/tex].

[Guest]

Topic: The Existence and the Number of Integral Lattice Points (X) on a Hypersurface (k) such that T(X) = k.

Reference Link: https://www.math10.com/forum/viewtopic.php?f=63&t=9719.

We have an outline for a good theory. We have much more work to do. Good luck!

[Guest]

Topic: The Existence and the Number of Integral Lattice Points (X) on a Hypersurface (k) such that T(X) = k.

Reference Link: https://www.math10.com/forum/viewtopic.php?f=63&t=9719.

We have an outline for a good theory. We have much more work to do. Good luck!

The idea of lattice points may not be appropriate here. But it's an elegant idea, and we will keep it until we find it is inappropriate.

[Guest]

Finding integral solutions to T(X) = k is generally like finding needles in a haystack. It is very challenging. And it would be very nice/very optimistic if the integral solutions (if they exist) of T(X)= k have some structure (algebraic, geometrical, or whatever).

[Guest]

Update: ... [tex]-282924 < x_{20 } < 282924[/tex].