On the Hilbert's 8th and 10th Problems

On the Hilbert's 8th and 10th Problems

FYI: 'Hilbert Walked so the Clay Mathematics Institute Could Run. The problems shaping modern mathematics' by Dr. E. J. Lamb,

https://blogs.scientificamerican.com/roots-of-unity/hilbert-walked-so-the-clay-mathematics-institute-could-run/.

In her article, Dr. Lamb believes that Hilbert's 10th is "clearly" solved. And that the Riemann Hypothesis, the central subproblem of Hilbert's 8th Problem, is "clearly" open.

I strongly disagree with her that Hilbert's 10th Problem is "clearly solved" since speculation (random and undirected or unintelligent endless solution searches of some DEs via an algorithm) is not proof! And therefore, I strongly believe Hilbert's 10th Problem is clearly open and is currently being investigated.

Moreover, I strongly believe the Riemann Hypothesis is clearly closed since Re(z) = 1/2 is clearly optimum for all primes and for all nontrivial zeros (z or zeta zeros) of Riemann Zeta Function and since the nth prime exists if only if the nth zeta zero exists...

Clearly,

David Cole

https://www.researchgate.net/profile/David_Cole29
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Re: On the Hilbert's 8th and 10th Problems

Guest wrote:FYI: 'Hilbert Walked so the Clay Mathematics Institute Could Run. The problems shaping modern mathematics' by Dr. E. J. Lamb,

https://blogs.scientificamerican.com/roots-of-unity/hilbert-walked-so-the-clay-mathematics-institute-could-run/.

In her article, Dr. Lamb believes that Hilbert's 10th Problem is "clearly" solved. And that the Riemann Hypothesis, the central subproblem of Hilbert's 8th Problem, is "clearly" open.

I strongly disagree with her that Hilbert's 10th Problem is "clearly" solved since speculation (random and undirected or unintelligent endless solution searches of some DEs via an algorithm) is not proof! And therefore, I strongly believe Hilbert's 10th Problem is clearly open and is currently being investigated.

Moreover, I strongly believe the Riemann Hypothesis is clearly closed since Re(z) = 1/2 is clearly optimum for all primes and for all nontrivial zeros (z or zeta zeros) of Riemann Zeta Function and since the nth prime exists if only if the nth zeta zero exists...

Clearly,

David Cole

https://www.researchgate.net/profile/David_Cole29

Remark: DE means Diophantine equation. Re(z) means the real part of the complex zeta zero (z).
Guest

Re: On the Hilbert's 8th and 10th Problems

Hilbert's 10th problem is a very difficult problem. We, humans, I believe do not yet fully understand Diophantine equations (DEs) generally.

There are some DEs that do not have rational solutions, and there are some DEs that do have rational solutions. How does one distinguish between the two?

How many rational solutions can a potentially solvable DE have?

How can one devise a very clever and general algorithm to solve efficiently DEs that do have rational solutions or to determine efficiently that some DEs do not have rational solutions?

Hmm. A general theory of DEs is required before we can devise that clever and general algorithm to solve any DEs over rational numbers, affirmatively or negatively.

Robinson-Matiyasevich's proof of the unsolvability of Hilbert's 10th Problem is unacceptable. Their proof lacks a sound and general understanding of DEs. Furthermore, there's no general theory of DEs that supports their proof.

Dave.
Guest

Re: On the Hilbert's 8th and 10th Problems

FYI: "The central problem of Diophantine geometry is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, an equation in two variables defines a curve in the plane. More generally, an equation, or system of equations, in two or more variables defines a curve, a surface or some other such object in n-dimensional space. In Diophantine geometry, one asks whether there are any rational points (points all of whose coordinates are rationals) or integral points (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is if there are finitely or infinitely many rational points on a given curve (or surface)."

-- 'Diophantine geometry', https://www.wikiwand.com/en/Number_theory#/Diophantine_geometry.

'Diophantine Geometry', https://www.wikiwand.com/en/Diophantine_geometry.
Guest

Re: On the Hilbert's 8th and 10th Problems

FYI: 'Hilbert Walked so the Clay Mathematics Institute Could Run. The problems shaping modern mathematics' by Dr. E. J. Lamb,

https://blogs.scientificamerican.com/roots-of-unity/hilbert-walked-so-the-clay-mathematics-institute-could-run/.

In her article, Dr. Lamb believes that Hilbert's 10th Problem is "clearly" solved. And that the Riemann Hypothesis, the central subproblem of Hilbert's 8th Problem, is "clearly" open.

I strongly disagree with her that Hilbert's 10th Problem is "clearly" solved since speculation (random and undirected or unintelligent endless solution searches of some DEs via an algorithm) is not proof! And therefore, I strongly believe Hilbert's 10th Problem is clearly open and is currently being investigated.

Moreover, I strongly believe the Riemann Hypothesis is clearly closed since Re(z) = 1/2 is clearly optimum for all primes and for all nontrivial zeros (z or zeta zeros) of Riemann Zeta Function and since the nth prime exists if only if the nth zeta zero exists...

Clearly,

David Cole ...

Remark: DE means Diophantine equation. Re(z) means the real part of the complex zeta zero (z).

Moreover, the follow-up story or essay,

'The Surprising Link between Recreational Math and Undecidability:
The Fibonacci numbers and Hilbert’s 10th problem

by Dr. E. J. Lamb is not clear!

It is not clear to us that there's a negative solution to Hilbert's Tenth Problem.
We wished that Dr. E. J. Lamb would be more explicit with regards to a specific Diophantine equation (DE) involving Fibonacci numbers or whatever that demonstrates that no algorithm can solve affirmatively or negatively over rational numbers.
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