What types of Diophantine equations are unsolvable?

[Guest]

We are stuck since the use of randomness can take us only so far unless we are extremely lucky!

There could be many or all hypersurfaces in [tex]k_{below } \le k \le k_{above}[/tex] that do not have integral solutions for T(X) in that range

[Guest]

We are stuck since the use of randomness can take us only so far unless we are extremely lucky!

There could be many or all hypersurfaces in [tex]k_{below } \le k \le k_{above}[/tex] that do not have integral solutions for T(X) in that range.

We should keep experimenting with the min and max values of the components of X while being mindful of the complicated bounds allowed/constrained by T(X) = k.

Again, the need for good theory is important...

[Guest]

We believe David Hilbert's Tenth Problem is closed!

Go Blue!

[Guest]

Hmm. Let's test our theory/algorithm. Please solve a polynomial DE with 100 unknown variables.

Please see attached file for all details. Good Luck!

[Guest]

Oops! The attached pdf of previous post does not have the variable, [tex]x_{40 }[/tex], in our equation. Please add [tex]-155x_{40 }[/tex] to our equation. Thanks! Good Luck!

[Guest]

Hint: [tex]-10 \le x_{i } \le 10[/tex] for some [tex]x_{i } \in X[/tex].

Does the equation, T(X) = k, have an integral solution?

Please see polyDE100.pdf for the details of equation.

[Guest]

Hmm. Understanding the details of a complicated polynomial Diophantine equation is important...

[Guest]

We believe David Hilbert's Tenth Problem is closed!

Go Blue!

Oops! The problem is open! There's much to learn...

[Guest]

We believe David Hilbert's Tenth Problem is closed!

Go Blue!

Oops! The problem is open! There's much to learn...

It's time for serious bookkeeping analysis of the details of our equation. And we recommend a spreadsheet to track the relations/values of all components of X for T(X) = k.

Some things to consider are the sign, magnitude, powers, inverse/direct proportionalities, etc. of all components/variables of X. We have the data. Right?

Let's do the work. Good luck!

[Guest]

FYI: We generate our data from our equation (see polyDE100.pdf for details of T(X) where

k = 8 472 827 925 723 497 279 079 825 725 896 743 212 537 092 151 545 454 399 977 711 146 464 646 464 645 272 151 245 423 102 640 450 424 554 210 121 024 221 004 240 012 421 214)

by randomizing the integral vector, X, such that

T(X) - k < t * k where t = 0.1, .01, .001, etc.

We are now aware of the limits of this process, but we will our best to gather sufficient data for a detailed analysis via a spreadsheet.

Good luck!

[Guest]

FYI: We generate our data from our equation (see polyDE100.pdf for details of T(X) where

k = 8 472 827 925 723 497 279 079 825 725 896 743 212 537 092 151 545 454 399 977 711 146 464 646 464 645 272 151 245 423 102 640 450 424 554 210 121 024 221 004 240 012 421 214)

by randomizing the integral vector, X, such that

T(X) - k < t * k where t = 0.1, .01, .001, etc.

We are now aware of the limits of this process, but we will our best to gather sufficient data for a detailed analysis via a spreadsheet.

Good luck!

Please remember k is king, and the relations of its many partitions as defined by X and T(X) is a measure of its kingdom (complexity or meaning). And there's also a great deal of poetry (wonder and beauty) to be discovered in the study of T(X) = k.

Enjoy that meditation!

[Guest]

Have we constructed a sound halting criterion for Hilbert's Tenth Problem?

[Guest]

Have we constructed a sound halting criterion for Hilbert's Tenth Problem?

Who knows?

My number theory adventures are now behind me...

And I have decided to work on David Hilbert's Sixth Problem or related research in order to advance the theory of energy.

But Hilbert's Tenth Problem, I strongly believe, is solvable (positive answer with a polynomial time algorithm). Am I wrong?

Dave.

"The further a mathematical
theory is developed, the more
harmoniously and uniformly
does its construction proceed,
and unsuspected relations are
disclosed between hitherto
separated branches of the
science." -- David Hilbert.