Review the Proof of David Hilbert's Tenth Problem

[Guest]

FYI: "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." --

https://en.m.wikipedia.org/wiki/Hilbert%27s_tenth_problem

My tentative view is that we, humans, are not yet clever enough to devise a general algorithm to solve Hilbert's Tenth Problem. There is a current 'proof' which states the problem is insolvable. We must challenge the validity of this grand result. The 'proof' may be seriously flawed...

Dave,

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

[Guest]

FYI: "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." --

https://en.m.wikipedia.org/wiki/Hilbert%27s_tenth_problem

My tentative view is that we, humans, are not yet clever enough to devise a general algorithm to solve Hilbert's Tenth Problem. There is a current 'proof' of the Matiyasevich's Theorem which states the problem is insolvable. We must challenge the validity of this grand and important result. The 'proof' may be seriously flawed...

Dave,

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

"Matiyasevich's Theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says:

Every computably enumerable set is Diophantine.
A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0.

Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0. Now we make an algorithm which simply tries all possible values for n, x1, ..., xk (in, say, some simple order consistent with the increasing order of the sum of their absolute values), and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk.

Proof technique:

Yuri Matiyasevich utilized a method involving Fibonacci numbers, which grow exponentially, in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam – hence, MRDP – had shown that this suffices to show that every computably enumerable set is Diophantine."

[Guest]

"Matiyasevich's Theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says:

Every computably enumerable set is Diophantine.
A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0.

Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0. Now we make an algorithm which simply tries all possible values for n, x1, ..., xk (in, say, some simple order consistent with the increasing order of the sum of their absolute values), and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk.

Proof technique:

Yuri Matiyasevich utilized a method involving Fibonacci numbers, which grow exponentially, in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam – hence, MRDP – had shown that this suffices to show that every computably enumerable set is Diophantine."

Relevant Reference Link:

'The MRDP Theorem - Logic Matters' --a pdf file,

http://www.math.umd.edu/~laskow/Pubs/713/Diophantine.pdf

[Guest]

The proof of the Matiyasevich's Theorem should be reconfirmed as sound... It's a worthwhile endeavor to reconfirm the proof since we may gain valuable insights on how to formulate unsolvable Diophantine equations which aid in the critical development of uncrackable ciphers employed to secure data transmissions over worldwide Internet...

Dave.

[Guest]

The proof of the Matiyasevich's Theorem should be reconfirmed as sound... It's a worthwhile endeavor to reconfirm the proof since we may gain valuable insights on how to formulate unsolvable Diophantine equations which aid in the critical development of uncrackable ciphers employed to secure data transmissions over worldwide Internet...

Dave.

Is the proof of the Matiyasevich's Theorem wrong?

Dave.

[Guest]

Reference:

'Proof as a Tool for Learning Mathematics',

http://web.mit.edu/bskow/www/215-S12/knuth_proof-as-a-tool-for-learning.pdf

[Guest]

The proof of the Matiyasevich's Theorem should be reconfirmed as sound... It's a worthwhile endeavor to reconfirm the proof since we may gain valuable insights on how to formulate unsolvable Diophantine equations which aid in the critical development of uncrackable ciphers employed to secure data transmissions over worldwide Internet...

Dave.

Is the proof of Matiyasevich's Theorem wrong?

Dave.

The 'proof' of Matiyasevich's Theorem seems not convincing enough, and is probably seriously flawed too.

Dave.

[Guest]

The proof of the Matiyasevich's Theorem should be reconfirmed as sound... It's a worthwhile endeavor to reconfirm the proof since we may gain valuable insights on how to formulate unsolvable Diophantine equations which aid in the critical development of uncrackable ciphers employed to secure data transmissions over worldwide Internet...

Dave.

Is the proof of Matiyasevich's Theorem wrong?

Dave.

The 'proof' of Matiyasevich's Theorem seems not convincing enough, and is probably seriously flawed too.

Dave.

"...algorithm which simply tries all possible values for n, x1, ..., xk (in, say, some simple order consistent with the increasing order of the sum of their absolute values), and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk."

We would like an example of an algorithm and a Diophantine equation, if possible, to demonstrate how the search for solutions to that Diophantine equation never halts or becomes an endless search.
Speculation is not proof!

[Guest]

David Hilbert's Tenth Problem is still open!

Good luck,

Dave.

[Guest]

We have a growing knowledge of the nature of Diophantine equations along with current/future methods to solve them efficiently. And because of our efforts, I believe David Hilbert's Tenth Problem is still open.

Dave.

[Guest]

"...algorithm which simply tries all possible values for n, x1, ..., xk (in, say, some simple order consistent with the increasing order of the sum of their absolute values), and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk."

We would like an example of an algorithm and a Diophantine equation, if possible, to demonstrate how an intelligently directed search for solutions to that Diophantine equation never halts or becomes an endless search.

Speculation (random and undirected searches) is not proof!

Mathematical methods of algorithms that search intelligently and efficiently for solutions of Diophantine equations should eliminate the possibility of endless search.

Dave.

[Guest]

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]

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.

Relevant Reference Link:

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

[Guest]

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' (https://blogs.scientificamerican.com/roots-of-unity/the-surprising-link-between-recreational-math-and-undecidability/)

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.

[Klais]

Yes, it is still relevant today.