What types of Diophantine equations are unsolvable?

[Guest]

Correction: The negative answer for Hilbert's Tenth Problem may be inapplicable here.

However, we are still skeptical about the negative answer for Hilbert's Tenth Problem.

[Guest]

Keywords/Key Ideas: Hilbert's Tenth Problem, Halting Problem, Halting Criteria, Newton's Method, Convergent Sequence, and Algorithm, etc.

Remark: Now we need a better theory!

[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.

An Attempt To Generate Solutions over [tex]-1 < \beta < 0[/tex]:

Let [tex]s \approx \beta + 1[/tex] < 1 such that [tex]\frac{234,567,890,123}{\beta + 1} \approx \ (\lambda_{approx} * 234,567,890,123)^{ \frac{1}{7} } \in \mathbb{Z}[/tex] for some [tex]\lambda_{approx} > 1[/tex].

We must compute [tex]\lambda_{true}[/tex] and [tex]\beta[/tex] based on [tex]s[/tex] such that [tex](\lambda_{true} * 234,567,890,123)^{ \frac{1}{7} } \in \mathbb{Z}[/tex].

[Guest]

An Update:

Let [tex]s \approx \beta + 1[/tex] < 1 such that [tex](\frac{234,567,890,123}{\beta + 1})^{ \frac{1}{7} } \approx \ (\lambda_{approx} * 234,567,890,123)^{ \frac{1}{7} } \in \mathbb{Z}[/tex] for some [tex]\lambda_{approx} > 1[/tex].

We must compute [tex]\lambda_{true}[/tex] and [tex]\beta[/tex] based on [tex]s[/tex] such that [tex](\lambda_{true} * 234,567,890,123)^{ \frac{1}{7} } \in \mathbb{Z}[/tex].

[Guest]

Remark: We believe we can construct a polynomial-time algorithm that answers our question.

[Guest]

Important Remark: [tex]s - q[/tex] for some [tex]q \in \mathbb{Q}[/tex] is an irreducible polynomial over the set of rational numbers, and therefore, our question has a negative answer (no integer solutions). Right?

Thus,[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{Q}[/tex] is unsolvable for [tex]x, y \in \mathbb{Z}[/tex] and for

[tex]x, y \in \mathbb{Q}[/tex]. Right?

That's the power and clarity of good theory!

[Guest]

Important Remark: [tex]s - q[/tex] for some [tex]q \in \mathbb{Q}[/tex] is an irreducible polynomial over the set of rational numbers, and therefore, our question has a negative answer (no integer solutions). Right?

Thus, [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{Q}[/tex] is unsolvable for [tex]x, y \in \mathbb{Z}[/tex] and for

[tex]x, y \in \mathbb{Q}[/tex]. Right?

That's the power and clarity of good theory!

If we are correct, we took too much time and effort to discover our result.

"Hmm. What simple is ..., is what simple does!"

[Guest]

Remark: If our result holds, we apologize for our stupidity... Thank you!

[Guest]

An Update:

Remark: [tex]s - q[/tex] for some [tex]q \in \mathbb{Q}[/tex] is an irreducible polynomial over the set of rational numbers, and therefore, our specific question has a negative answer (no integer solutions). Right?

Thus, [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{Q}[/tex] may be unsolvable for [tex]x, y \in \mathbb{Z}[/tex] and for

[tex]x, y \in \mathbb{Q}[/tex]. BUT THERE ARE EXCEPTIONS GALORE!


For example, let [tex]x = 8[/tex] and let [tex]y = -19[/tex], ....

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Remark: Our work on this problem is not done.

[Guest]

An Update:

Remark: [tex]s - q[/tex] for some [tex]q \in \mathbb{Q}[/tex] is an irreducible polynomial over the set of rational numbers, and therefore, our specific question has a negative answer (no integer solutions). Right?

Thus, [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{Q}[/tex] may be unsolvable for [tex]x, y \in \mathbb{Z}[/tex] and for

[tex]x, y \in \mathbb{Q}[/tex]. BUT THERE ARE EXCEPTIONS GALORE!


For example, let [tex]x = 8[/tex] and let [tex]y = -19[/tex], ....

Are there just three exceptions (existence of possible solutions...)? And if there are three exceptions, what are they?

Hint: The range of the parameter, [tex]\beta[/tex], is important.

Please ask the experts since they have the answers.

[Guest]

Oops! For our specific example, [tex]c = 234,567,890,123[/tex] should be [tex]c = 234,567,890,122[/tex] or some other even integer. Why?

[Guest]

FYI: 'On Hilbert's Tenth Problem' by Prof. Y. Matiyasevich,

https://mathtube.org/sites/default/files/lecture-notes/Matiyasevich.pdf.

[Guest]

Remark: We are not satisfied with our work. We need a solution (affirmative or negative) for our problems... The work will continue.

[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} =234,567,890,122[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let [tex]s(\beta) = \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{234,567,890,122}{s( \beta )})^{ \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?

Since [tex](234,567,890,123)^{ \frac{1}{7} } < 43[/tex], we have [tex]s( \beta ) - \frac{234,567,890,122}{ \lambda^{7} } = 0[/tex] for some [tex]\lambda \in \mathbb{Z}[/tex] such that [tex]\lambda \ge 43[/tex].

However, we claim [tex]s( \beta ) - \frac{234,567,890,122}{ \lambda^{7} } = 0[/tex] is unsolvable for [tex]\beta \in \mathbb{Q}[/tex] such that [tex]-1 < \beta < 0[/tex].

Moreover, we claim [tex]s( \beta ) - \frac{2c}{ \lambda^{7} } = 0[/tex] is unsolvable for [tex]\beta \in \mathbb{Q}[/tex] such that [tex]-1 < \beta < 0[/tex] and [tex]\frac{2c}{ \lambda^{7}} \in \mathbb{Q}[/tex] such that [tex]0 < \frac{2c}{ \lambda^{7}} < 1[/tex].

Thus, if we are correct, then our specific example and our general example have negative answers.

[Guest]

Remark: If our claims are proven true, then we are satisfied with our work.

[Guest]

Remark: We assume [tex]2c \in 2 \mathbb{Z}[/tex]...

[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} =234,567,890,122[/tex] for some [tex]x \ne y \in \mathbb{Z}[/tex]?

Let [tex]s(\beta) = \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{234,567,890,122}{s( \beta )})^{ \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?

Since [tex](234,567,890,123)^{ \frac{1}{7} } < 43[/tex], we have [tex]s( \beta ) - \frac{234,567,890,122}{ \lambda^{7} } = 0[/tex] for some [tex]\lambda \in \mathbb{Z}[/tex] such that [tex]\lambda \ge 43[/tex].

However, we claim [tex]s( \beta ) - \frac{234,567,890,122}{ \lambda^{7} } = 0[/tex] is unsolvable for [tex]\beta \in \mathbb{Q}[/tex] such that [tex]-1 < \beta < 0[/tex].

VERY FLAWED CLAIM: Moreover, we claim [tex]s( \beta ) - \frac{2c}{ \lambda^{7} } = 0[/tex] is unsolvable for [tex]\beta \in \mathbb{Q}[/tex] such that [tex]-1 < \beta < 0[/tex] and [tex]\frac{2c}{ \lambda^{7}} \in \mathbb{Q}[/tex] such that [tex]0 < \frac{2c}{ \lambda^{7}} < 1[/tex].

Thus, if we are correct, then our specific example has a negative answer. And our general claim may be solved depending on the value of c.

Remark: Our work is not done since we do not know the truth...

[Guest]

Oops! We apologize for the mess we have created.

[Guest]

Oops! We apologize for the mess we have created.

The best is yet to come. Please keep working. Good luck too!