What types of Diophantine equations are unsolvable?

[Guest]

Halting Criteria Attempt (adjusting the d-parameter appropriately):

Let [tex]d = |x - y| \approx ( \frac{\beta - 1}{\beta} ) (\frac{c}{\beta - 1})^{ \frac{1}{7} }[/tex] where [tex]0 < \beta \in \mathbb{Q_{+}}<1[/tex] for some [tex]c \in \mathbb{2Z_{+}}[/tex].

[Guest]

Halting Criteria Attempt (adjusting the d-parameter appropriately):

Let [tex]d = |x - y| \approx ( \frac{\beta - 1}{\beta} ) (\frac{c}{\beta - 1})^{ \frac{1}{7} }[/tex] where [tex]0 < \beta \in \mathbb{Q_{+}}<1[/tex] for some [tex]c \in \mathbb{2Z_{+}}[/tex].

Since [tex]y \approx \frac{d}{\beta - 1}[/tex], [tex]d = |x - y| \approx (\beta - 1 ) (\frac{c}{\beta - 1})^{ \frac{1}{7} }[/tex]...

It is all tentative and maybe not error-free too.

[Guest]

Update:

Since [tex]y \approx \frac{d}{1+ \beta}[/tex], [tex]d = |x - y| \approx |( \beta -1 ) (\frac{c}{\beta + 1 })^{ \frac{1}{7} }|[/tex] where [tex]-1 < \beta \in \mathbb{Q_{-}}< 0[/tex] for

some [tex]c \in \mathbb{2Z_{+}}[/tex].

Example: For [tex]c = 234,567,890,122[/tex] and [tex]\beta = -.5[/tex],

we have [tex]d \approx ( 1.5) (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } \approx 69[/tex];

where [tex]y \approx (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } = 46[/tex]

and [tex]x \approx -23[/tex].

Recall: [tex]s = \sum_{i=1}^{7}\beta{i} + 1 = 1 -0.3359375 = < \approx \beta + 1 = .5[/tex].

So [tex]y \approx (\frac{ 234,567,890,122}{0.664062})^{ \frac{1}{7} } = 44[/tex]

and [tex]x \approx -22[/tex].

Therefore, [tex]\sum_{n=7}^{0 }(44)^{7-n}(-22)^n = 212,020,420,480 < c =234,567,890,122[/tex] when [tex]\beta = -.5[/tex].

[Guest]

Update:

Since [tex]y \approx \frac{d}{1+ \beta}[/tex], [tex]d = |x - y| \approx |( \beta -1 ) (\frac{c}{\beta + 1 })^{ \frac{1}{7} }|[/tex] where [tex]-1 < \beta \in \mathbb{Q_{-}}< 0[/tex] for

some [tex]c \in \mathbb{2Z_{+}}[/tex].

Example: For [tex]c = 234,567,890,122[/tex] and [tex]\beta = -.5[/tex],

we have [tex]d \approx ( 1.5) (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } \approx 69[/tex];

where [tex]y \approx (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } = 46[/tex]

and [tex]x \approx -23[/tex].

Recall: [tex]s = \sum_{i=1}^{7}\beta{i} + 1 = 1 -0.3359375 = 0.664062 > \beta + 1 = .5[/tex].

So [tex]y \approx (\frac{ 234,567,890,122}{0.664062})^{ \frac{1}{7} } = 44[/tex]

and [tex]x \approx -22[/tex].

Therefore, [tex]\sum_{n=7}^{0 }(44)^{7-n}(-22)^n = 212,020,420,480 < c =234,567,890,122[/tex] when [tex]\beta = -.5[/tex].

Further computations indicate

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} \ne 234,567,890,122[/tex] for [tex]x, y \in \mathbb{Z}[/tex].

[Guest]

Update:

Since [tex]y \approx \frac{d}{1+ \beta}[/tex], [tex]d = |x - y| \approx |( \beta -1 ) (\frac{c}{\beta + 1 })^{ \frac{1}{7} }|[/tex] where [tex]-1 < \beta \in \mathbb{Q_{-}}< 0[/tex] for

some [tex]c \in \mathbb{2Z_{+}}[/tex].

Example: For [tex]c = 234,567,890,122[/tex] and [tex]\beta = -.5[/tex],

we have [tex]d \approx ( 1.5) (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } \approx 69[/tex];

where [tex]y \approx (\frac{ 234,567,890,122}{1- .5 })^{ \frac{1}{7} } = 46[/tex]

and [tex]x \approx -23[/tex].

Recall: [tex]s = \sum_{i=1}^{7}\beta{i} + 1 = 1 -0.3359375 = 0.664062 > \beta + 1 = .5[/tex].

So [tex]y \approx (\frac{ 234,567,890,122}{0.664062})^{ \frac{1}{7} } = 44[/tex]

and [tex]x \approx -22[/tex].

Therefore, [tex]\sum_{n=7}^{0 }(44)^{7-n}(-22)^n = 212,020,420,480 < c =234,567,890,122[/tex] when [tex]\beta = -.5[/tex].

Further computations indicate

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} \ne 234,567,890,122[/tex] for [tex]x, y \in \mathbb{Z}[/tex].

Moreover, we can determine the solvability or the unsolvability of

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in 2\mathbb{Z_{+}}[/tex].

[Guest]

Remark: We apologize for the sloppy (flawed) work of some previous posts.

But we are satisfied with the final result.

[Guest]

Remark: Our work indicates there is only one unique solution for a given c value (if the solution exists) of the following Diophantine equation,

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in 2\mathbb{Z_{+}}[/tex].

[Guest]

Remark: Our work indicates there is only one unique solution for a given c value (if the solution exists) of the following Diophantine equation,

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in 2\mathbb{Z_{+}}[/tex].

In general, we can determine the solutions (if they exist) of

[tex]\sum_{n=0}^{k}x^{k-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in \mathbb{Z}[/tex] and [tex]k \in \mathbb{Z_{+}}[/tex].

[Guest]

Remark: Our work indicates there is only one unique solution for a given c value (if the solution exists) of the following Diophantine equation,

[tex]\sum_{n=0}^{7 }x^{7-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in 2\mathbb{Z_{+}}[/tex].

In general, we can determine the solutions (if they exist) of

[tex]\sum_{n=0}^{k}x^{k-n}y^{n} = c[/tex] for [tex]x, y \in \mathbb{Z}[/tex] where [tex]c \in \mathbb{Z}[/tex] and [tex]k \in \mathbb{Z_{+}}[/tex].

Remark: Two important equations, [tex]x = \beta y[/tex] and [tex]d = |x - y|[/tex], help us to establish a halting criteria for our algorithm.

[Guest]

Important Remark: We declare David Hilbert's Tenth Problem (https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) as an open problem...

And our work will continue...

Dave.

Go Blue!

[Guest]

Important Remark: We declare David Hilbert's Tenth Problem (https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) as an open problem...

And our work will continue...

Dave.

Go Blue!

P.S. Hilbert's Tenth Problem is a great problem, and we hope we can construct a grand algorithm to solve it or discover an unsolvable Diophantine equation along the way...

Good Luck!

[Guest]

Remark: Hmm. It may be easier to discover an unsolvable Diophantine equation that may have or may not have a solution but defeats any algorithm to compute the true result than to construct a grand algorithm that solves Hilbert's Tenth Problem.

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

[Guest]

Remark: Hmm. It may be easier to discover an unsolvable Diophantine equation that may have or may not have a solution but defeats any algorithm to compute the true result than to construct a grand algorithm that solves Hilbert's Tenth Problem.

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

The MRDP theorem may help us to discover that unsolvable Diophantine equation. It's worth a try. Good luck!

[Guest]

Reference Links: 'Recursively Enumerable Sets',

https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem#Recursively_enumerable_sets;

'Chapter 4 A Universal Program 1 Coding programs',

https://slidetodoc.com/chapter-4-a-universal-program-1-coding-programs/.

[Guest]

Update: ...

Recall: [tex]s = \sum_{i=1}^{7}\beta^{i} + 1 = 1 -0.3359375 = 0.664062 > \beta + 1 = .5[/tex]...

[Guest]

Remark: MRDP (Matiyasevich's ) theorem is difficult to understand and seems doubtful too. But we could be wrong. We will continue to work on it.

Meanwhile, let's solve the following Diophantine equation,

[tex]x^{3} + 4x^2z -8xy^{2}-44xyz^{3}-9z^{3} = 8,170,698,744,133,625,080,294,398[/tex] for [tex]x,y, z \in \mathbb{Z}[/tex].

Hint: The solution does exist. Good luck!

[Guest]

Remark: MRDP (Matiyasevich's ) theorem is difficult to understand and seems doubtful too. But we could be wrong. We will continue to work on it.

Meanwhile, let's solve the following Diophantine equation,

[tex]x^{3} + 4x^2z -8xy^{2}-44xyz^{3}-9z^{3} = 8,170,698,744,133,625,080,294,398[/tex] for [tex]x,y, z \in \mathbb{Z}[/tex].

Hint: The solution does exist. Good luck!

Let [tex]x = \beta_{1} y[/tex] and [tex]d_{xy }= |x - y|[/tex];

And let [tex]z = \beta_{2} y[/tex] and [tex]d_{zy }= |z- y|[/tex] for some [tex]\beta_{1}, \beta_{2} \in \mathbb{Q}[/tex]...

Have fun!

[Guest]

[tex]x^{3} + 4x^2z -8xy^{2}-44xyz^{3}-9z^{3}

= (\beta_{1} y)^{3} + 4(\beta_{1} y)^2( \beta_{2} y) -8(\beta_{1} y)y^{2}-44(\beta_{1} y)y( \beta_{2} y)^{3}-9( \beta_{2} y)^{3}

= (\beta_{1}^{3} + 4\beta_{1}^{2} \beta_{2} -8\beta_{1} - 9 \beta_{2}^{3})y^{3} - 44 \beta_{1} \beta_{2}^{3}y^{5} = 8,170,698,744,133,625,080,294,398[/tex].

Hmm. What is the 'Rabbit Hole' analysis for [tex]\beta_{1}[/tex] and [tex]\beta_{2}[/tex]?

And can we estimate roughly the value of [tex]y[/tex]?

[Guest]

[tex]x^{3} + 4x^2z -8xy^{2}-44xyz^{3}-9z^{3}

= (\beta_{1} y)^{3} + 4(\beta_{1} y)^2( \beta_{2} y) -8(\beta_{1} y)y^{2}-44(\beta_{1} y)y( \beta_{2} y)^{3}-9( \beta_{2} y)^{3}

= (\beta_{1}^{3} + 4\beta_{1}^{2} \beta_{2} -8\beta_{1} - 9 \beta_{2}^{3})y^{3} - 44 \beta_{1} \beta_{2}^{3}y^{5} = 8,170,698,744,133,625,080,294,398[/tex].

Hmm. Q1: What is the 'Rabbit Hole' analysis for [tex]\beta_{1}[/tex] and [tex]\beta_{2}[/tex]?

Q2: And can we estimate roughly the value of [tex]y[/tex]?

On Q2: [tex]y(\beta_{1}^{3} + 4\beta_{1}^{2} \beta_{2} -8\beta_{1} - 9 \beta_{2}^{3}) - 44 \beta_{1} \beta_{2}^{3}y^{2})^{1/3} \approx 201,412,490.4[/tex] ...

[Guest]

Update:

On Q2: [tex]y(\beta_{1}^{3} + 4\beta_{1}^{2} \beta_{2} -8\beta_{1} - 9 \beta_{2}^{3} - 44 \beta_{1} \beta_{2}^{3}y^{2})^{1/3} \approx 201,412,490.4[/tex] ...

Remark: We need to answer Q1 before we can answer Q2. Right?