What types of Diophantine equations are unsolvable?

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Question: What types of Diophantine equations are unsolvable over the set of positive rational numbers?

Reference Link: https://math.stackexchange.com/questions/4243310/what-types-of-diophantine-equations-are-not-solvable-over-the-set-of-positive-ra

[Guest]

FYI: 'From the result of Hilbert's tenth problem, does it follow that there exist infinitely numbers of equation...',

https://math.stackexchange.com/questions/118662/from-the-result-of-hilberts-tenth-problem-does-it-follow-that-there-exist-infi?rq=1.

[Guest]

Good Reference Book: 'An Introduction to Diophantine Equations' by Prof. T. Andreescu et al.

https://www.springer.com/gp/book/9780817645489.

[Guest]

Question: What types of Diophantine equations are unsolvable over the set of positive rational numbers?

Reference Link: https://math.stackexchange.com/questions/4243310/what-types-of-diophantine-equations-are-not-solvable-over-the-set-of-positive-ra

I had a very bad experience with https://math.stackexchange.com, and I strongly do not recommend it to anyone.

Dave.

[Guest]

A General Example:

Can we generally determine the solvability or the unsolvability for the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} = c \in \mathbb{Z_{+}}[/tex]?

Remark: [tex]x, y \in \mathbb{Z}[/tex].

Hmm. The value of c is important!


A Specific Example:

Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex]?

[Guest]

A General Example:

Can we generally determine the solvability or the unsolvability for the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} = c \in \mathbb{Z_{+}}[/tex]?

Remark: [tex]x, y \in \mathbb{Z}[/tex].

Hmm. The value of c is important!


A Specific Example:

Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex]?

If we are unhappy with [tex]x, y \in \mathbb{Z}[/tex], then we could choose [tex]x, y \in \mathbb{Q}[/tex].

It is still a fair game.

[Guest]

Remark: We need to develop a theory that works well. Good luck!

[Guest]

Remark: The simplest case occurs when [tex]x = y[/tex]. Right?

We should assume [tex]x \ne y[/tex].

[Guest]

Some Food for Thought: 'NP-Complete decision problems for binary quadratics', by Profs., K. L. Manders and L. Adleman,

https://www.sciencedirect.com/science/article/pii/0022000078900442?via%3Dihub.

[Guest]

Wow! A single number has so much meaning/richness attached to it.

And the study of numbers tells us or reveals to us the truth about the nature of existence/things (energy, etc.)

Why? (A Hint: https://www.math10.com/forum/viewtopic.php?f=63&t=8579).

But we must be energetic, bold, resourceful, and imaginative to know their hidden/subtle/powerful secrets.

The great Gauss was right to say that number theory is the queen of the queens (the sciences including its greatest science, mathematics). Amen!

[Guest]

"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let s = [tex]\sum_{i=1}^{7} \beta^{i} + 1[/tex] for some [tex]\beta \in \mathbb{Q}[/tex].

Question 1: Does [tex]s | 234,567,890,123[/tex] where [tex]\beta < 0[/tex] or [tex]0 <\beta < 1[/tex] or [tex]\beta >1[/tex]?

Question 2: Does [tex]y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}[/tex]?

Question 3: Does [tex]x = \beta y \in \mathbb{Z}[/tex]?

If our questions, one, two, and three, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

[Guest]

An Update:

"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let s = [tex]\sum_{i=1}^{7} \beta^{i} + 1[/tex] for some [tex]\beta \in \mathbb{Q}[/tex].

Question1: Does [tex]y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}[/tex] where [tex]\beta < 0[/tex] or [tex]0 <\beta < 1[/tex] or [tex]\beta >1[/tex]?

Question 2: Does [tex]x = \beta y \in \mathbb{Z}[/tex]?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

[Guest]

An Update:

"Simple seeks simplest (best) solution."

A Specific Example:

Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let s = [tex]\sum_{i=1}^{7} \beta^{i} + 1[/tex] for some [tex]\beta \in \mathbb{Q}[/tex].

Question 1: Does [tex]y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}[/tex] where [tex]\beta < 0[/tex] or [tex]0 <\beta < 1[/tex] or [tex]\beta >1[/tex]?

Question 2: Does [tex]x = \beta y \in \mathbb{Z}[/tex]?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

Important Question: What is the algorithm that answers our question in polynomial time or less than polynomial time?

[Guest]

Hmm. Current theory (Hilbert's Tenth Problem has a negative answer.) states that we cannot generally answer our question in polynomial time or less...

We are skeptical since we had not been clever enough in our search for truth.

[Guest]

[b]Remark[\b]: We need better theory!

[Guest]

Remark: We need better theory!

[Guest]

Do we choose Rabbit Hole A: [tex]\beta < 0[/tex]?

Or do we choose Rabbit Hole B: [tex]0 < \beta < 1[/tex]?

Or do we choose Rabbit Hole C: [tex]\beta > 1[/tex]?

Hmm. We are clueless.

[Guest]

Do we choose Rabbit Hole A: [tex]\beta < 0[/tex]?

Or do we choose Rabbit Hole B: [tex]0 < \beta < 1[/tex]?

Or do we choose Rabbit Hole C: [tex]\beta > 1[/tex]?

Hmm. We are clueless.

It is not a hopeless situation. Thank God! Amen!

Keywords: Initial values, Criteria, Newton's Method (Jacobian), Convergent sequence...

Can we select initial values, [tex]x_{0 }, y_{0 } \in \mathbb{Z}[/tex], based on some criteria (unknown) that will help us to choose the best Rabbit Hole to explore?

Equation 0: [tex]\beta_{0 } = \frac{y_{0 }}{x_{0 }} \in \mathbb{Q}[/tex] ...

Relevant Reference Link: 'Are Diophantine equations in four variables solvable?'

https://www.math10.com/forum/viewtopic.php?f=63&t=7803.

[Guest]

An Update:

Do we choose Rabbit Hole A: [tex]\beta < -1[/tex]?

Or do we choose Rabbit Hole B: [tex]-1 < \beta < 0[/tex]?

Or do we choose Rabbit Hole C: [tex]0 < \beta < 1[/tex]?

Or do we choose Rabbit Hole D: [tex]\beta > 1[/tex]?

Hmm. There may be no solutions for each Rabbit Hole. Or there may be solutions for each Rabbit Hole. How many solutions? Who knows?

But we believe there are finitely many solutions (if they exist) for Rabbit Holes, A, C, and D. Why?

Therefore, the best Rabbit Hole to explore for possible solutions is Rabbit Hole B since it has the most possible solutions. Why?

[Guest]

An Update:

"Simple seeks simplest (best) solution."

A Specific Example:

Question: Can we solve the following Diophantine equation,

[tex]x^{7} + x^{6}y + x^{5}y^{2} + x^{4}y^{3}+ x^{3}y^{4}+ x^{2}y^{5}+ xy^{6} + y^{7} =234,567,890,123[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let s = [tex]\sum_{i=1}^{7} \beta^{i} + 1[/tex] for some [tex]\beta \in \mathbb{Q}[/tex] such that [tex]-1 < \beta < 0[/tex].

Question 1: Does [tex]y = (\frac{234,567,890,123}{s})^{ \frac{1}{7} } \in \mathbb{Z}[/tex]?

Question 2: Does [tex]x = \beta y \in \mathbb{Z}[/tex]?

If our questions, one and two, have affirmative answers, then we can solve our equation, otherwise, we cannot solve it. Right?

What is the algorithm that answers our question in polynomial time or less than polynomial time?

Relevant Reference Link:

'Are Diophantine equations in four variables solvable?'

https://www.math10.com/forum/viewtopic.php?f=63&t=7803.