# On the Shapes of Surfaces and the Solutions to DEs

### On the Shapes of Surfaces and the Solutions to DEs

Some Food for Thought:

Geometry meets number theory.

Please ponder curved surfaces, flat surfaces, and boundaries of surfaces when seeking solutions to DEs (Diophantine Equations).

And please think big (beyond 3 dimensions).

Good luck!

Go Blue!
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: There may be a deep connection (theorem) between the shape of space (surfaces including boundaries, curvature, other metrics) and the solvability of DEs.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

A Point of Clarity: The shape of space is broadly determined by the function associated with a Diophantine equation.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: There are three possibilities.

1. DEs with integral solutions;

2. DEs without integral solutions;

3. DEs that cannot be determined either way (1 or 2).

Let a DE be described by T(X) = k. The shape of space is determined by T(X).

And how does the shape of space determine 1, 2, and 3?
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: We are clueless!
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Guest wrote:Remark: We are clueless!

Hah! We can do better, and we will do better!

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

FYI: 'Hilbert metric',

https://en.m.wikipedia.org/wiki/Hilbert_metric.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

"Everything should be made as simple as possible, but no simpler." -- Albert Einstein.

Remark: Analysis is vital...

And we are in a period of learning and pondering.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: Yes! Depending on k, T(X) filters most or all relevant X $$\in \mathbb{Z^{n}}$$. And we should be able to determine when it does.

Moreover, we are close to claiming that Hilbert's Tenth Problem has an affirmative answer, but we have not done the necessary computations yet. So we are not sure about our tentative claim.

Thank Lord GOD! Amen!

Go Blue! Thank you!
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: Hilbert's Tenth Problem is a difficult problem!
And it is also a great problem.

Good Luck!
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: We expect an answer to our problem before Hilbert's 160th birthday on January 23, 2022.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Why should T(X) filter most or all relevant X $$\in \mathbb{Z^{n}}$$?
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Guest wrote:Why should T(X) filter most or all relevant X $$\in \mathbb{Z^{n}}$$?

Are you asking why T(X) allows a few or no integral solutions to T(X) = k?
Guest

Yes!
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Guest wrote:
Guest wrote:Why should T(X) filter most or all relevant X $$\in \mathbb{Z^{n}}$$?

Easy Answer: $$card(\mathbb{Z^{n}}) << card(\mathbb{R^{n}})$$.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

... Better Answer: $$\frac{|\mathbb{Z^{n}}|}{| \mathbb{R^{n}} |} = 0$$
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Remark: We believe Hilbert metric is quite useful here.

But it is not clear to us on how we should proceed. We have much to learn and to consider.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

An Observation:

T(X) = 0 may have no solutions or a finite number of solutions.

T(X) = 1 may have no solutions or a finite number of solutions.

T(X) = 2 may have no solutions or a finite number of solutions.

...

Why?
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

Update:

Guest wrote:An Observation:

T(X) = 0 may have no integral solutions or a finite number of integral solutions.

T(X) = 1 may have no integral solutions or a finite number of integral solutions.

T(X) = 2 may have no integral solutions or a finite number of integral solutions.

...

Why?
Guest

### Re: On the Shape of Space and the Solutions to DEs

Hey, don't forget to consider the negative integers, k, when searching for integral solutions to the Diophantine equation, T(X) = k.
Guest

### Re: On the Shapes of Surfaces and the Solutions to DEs

There may be solutions galore, $$X \in \mathbb{R^{n}}$$, for the Diophantine equation, T(X) = k. But there could also be no solutions, $$X \in \mathbb{Z^{n}}$$, such that T(X) = k for n > 3.

Go figure!

Exhaustive Search:

We assume T(X) = k and $$X \in \mathbb{R^{n}}$$ such $$X \notin \mathbb{Z^{n}}$$ for n > 3.

Let $$X + \triangle X \ne X$$ such that T($$X + \triangle X$$) = k where $$\triangle X \in \mathbb{R^{n}}$$.

Does the sum, $$X + \triangle X \in \mathbb{Z^{n}}$$?

It's time to do some computing. Good Luck!

While there are some or all $$x_{i } \in X$$ that are not integers, we try to adjust $$\triangle x_{i } \in \triangle X \in \mathbb{R^{n}}$$ so that $$x_{i } + \triangle x_{i } \in \mathbb{Z^{n}}$$.

And me must solve T($$X + \triangle X$$) = k.

We repeat the process until $$X + \triangle X \in \mathbb{Z^{n}}$$.

Does the process determines a solution or no solution?
Guest

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