What types of Diophantine equations are unsolvable?

[Guest]

Remark: Our efforts are experimental mathematics.

[Guest]

Hmm. Solving for [tex]\beta_{1}[/tex] is unworkable. Right?

Can we estimate roughly the range of possible values for [tex]\beta_{1}[/tex] or [tex]\beta_{ 2}[/tex] by holding [tex]y[/tex] as a constant and by employing the Newton's Method?

It's worth a try.

We hold [tex]y[/tex] as a constant and vary the values of [tex]\beta_{ 2}[/tex] and determine the range of possible values for [tex]\beta_{1}[/tex].

Or we hold [tex]y[/tex] as a constant and vary the values of [tex]\beta_{ 1}[/tex] and determine the range of possible values for [tex]\beta_{2}[/tex].

Let's generate some results. Good Luck!

Remark: The process is workable, and we expect some good results too.

The equation,

[tex](\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] with

[tex]-96,039 < y < 96,039[/tex], helps us to achieve our goals.

[Guest]

Remark: Our latest efforts are workable, but we can do better. We must do better!

Imagine a Diophantine equation with twenty or more unknown variables. How do we solve it?

[Guest]

Remark: Our latest efforts are workable, but we can do better. We must do better!

Imagine a Diophantine equation with twenty or more unknown variables. How do we solve it?

Two Difficult Diophantine Equations with 20 Unknown Variables:

[tex](3x_{1 } - 9 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3
= -3,860,334,356,048,162,706,947,124,014,629,721,458,422,079,179,365,900,014,318,627,999,237,678,912,202,034,469,964,261,741,230,576,848[/tex]
*****

[tex](3x_{1 } - 9 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3

=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,613,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666[/tex]
*****
Remark: We will tackle them later.

[Guest]

Remark: Solving (generally) Diophantine equations is a serious 'bookkeeping task'. The numbers (integers) must add up correctly, or they don't. Either way, "we must know, and we will know!" (David Hilbert)

Moreover, our bookkeeping will probably require matrices and some advanced tools of algebra.

[Guest]

Remark: Solving (generally) Diophantine equations is a serious 'bookkeeping task'. The numbers (integers) must add up correctly, or they don't. Either way, "we must know, and we will know!" (David Hilbert)

Moreover, our bookkeeping will probably require matrices and some advanced tools of algebra.

[tex]T: \mathbb {Z^{n }} \rightarrow \mathbb {Z}[/tex] or T(X) = [tex]k \in \mathbb{Z }[/tex] where X = ([tex]x_{1 }[/tex],[tex]x_{2 }[/tex], ..., [tex]x_{n }[/tex]) [tex]\in \mathbb{Z^{n } }[/tex].

Tentatively, some key topics may include perturbation theory, eigenvalues and eigenvectors, etc. where [tex]x \in[/tex] {[tex]x_{1 }[/tex],[tex]x_{2 }[/tex], ..., [tex]x_{n }[/tex]} and [tex]\lambda_{i}[/tex] are pivotal.

[Guest]

Update (Changed Second Equation):

Two Difficult Diophantine Equations with 20 Unknown Variables:

[tex](3x_{1 } + 8 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3
= -3,860,334,356,048,162,706,947,124,014,629,721,458,422,079,179,365,900,014,318,627,999,237,678,912,202,034,469,964,261,741,230,576,848[/tex]
*****

[tex](-5x_{1 } - 9 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 }) (x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^3x_{18}x_{20}^3

=1,762,676,465,144,323,736,067,531,492,445,426,472,572,506,888,133,023,637,595,124,094,781,513,311,794,822,925,047,637,182,709,666[/tex]
*****

How do we solve them?

We are almost clueless.

We are in a period of learning and pondering.

Hints: First equation has an unknown solution. And we do not know the status of the second equation.

[Guest]

Remark: Solving (generally) Diophantine equations is a serious 'bookkeeping task'. The numbers (integers) must add up correctly, or they don't. Either way, "we must know, and we will know!" (David Hilbert)

Moreover, our bookkeeping will probably require matrices and some advanced tools of algebra.

[tex]T: \mathbb {Z^{n }} \rightarrow \mathbb {Z}[/tex] or T(X) = [tex]k \in \mathbb{Z }[/tex] where X = ([tex]x_{1 }[/tex],[tex]x_{2 }[/tex], ..., [tex]x_{n }[/tex]) [tex]\in \mathbb{Z^{n } }[/tex].

Tentatively, some key topics may include perturbation theory, eigenvalues and eigenvectors, etc. where [tex]x \in[/tex] {[tex]x_{1 }[/tex],[tex]x_{2 }[/tex], ..., [tex]x_{n }[/tex]} and [tex]\lambda_{i}[/tex] are pivotal.

[Guest]

Let X = ([tex]x_{1 }[/tex], [tex]x_{2 }[/tex], [tex]x_{3 }[/tex], [tex]x_{4 }[/tex], [tex]x_{5 }[/tex], [tex]x_{6 }[/tex], [tex]x_{7 }[/tex], [tex]x_{8 }[/tex], [tex]x_{9 }[/tex], [tex]x_{10 }[/tex], [tex]x_{11 }[/tex], [tex]x_{12 }[/tex], [tex]x_{13 }[/tex], [tex]x_{14 }[/tex], [tex]x_{15 }[/tex], [tex]x_{16 }[/tex], [tex]x_{17 }[/tex], [tex]x_{18 }[/tex], [tex]x_{19 }[/tex], [tex]x_{20 }[/tex]).

And we let

T(X) = [tex](3x_{1 } + 8 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 })(x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3[/tex].

We must compute the following partial derivatives of T(X). (Why? We need to know how T(X) generally behaves.)

[tex]\frac{\partial T}{\partial x_{1 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{2 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{3} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{4 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{5 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{6} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{7} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{8 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{9 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{10 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{11} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{12 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{13 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{14 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{15} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{16 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{17 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{18 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{19} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{20 } } = ?[/tex]

[Guest]

Let X = ([tex]x_{1 }[/tex], [tex]x_{2 }[/tex], [tex]x_{3 }[/tex], [tex]x_{4 }[/tex], [tex]x_{5 }[/tex], [tex]x_{6 }[/tex], [tex]x_{7 }[/tex], [tex]x_{8 }[/tex], [tex]x_{9 }[/tex], [tex]x_{10 }[/tex], [tex]x_{11 }[/tex], [tex]x_{12 }[/tex], [tex]x_{13 }[/tex], [tex]x_{14 }[/tex], [tex]x_{15 }[/tex], [tex]x_{16 }[/tex], [tex]x_{17 }[/tex], [tex]x_{18 }[/tex], [tex]x_{19 }[/tex], [tex]x_{20 }[/tex]).

And we let

T(X) = [tex](3x_{1 } + 8 x_{5 }x_{6 } ) ( 14x_{3 }^3 - 56 x_{4 } x_{7 }) (2x_{4 } - 9x_{2 }^2 x_{9 }) (17x_{6 } - 61 x_{8} x_{11 }) ( x_{13} + x_{16 }^2-9 x_{14 } x_{18 }) ( x_{12 } - 9 x_{15 } x_{19 })(x_{10 } - 9 x_{17 } x_{20 })-4,034,500,221x_{1}^2x_{4}x_{7}^3x_{9}x_{10}^2x_{18}x_{20}^3[/tex].

We must compute the following partial derivatives of T(X). (Why? We need to know how T(X) generally behaves.)

[tex]\frac{\partial T}{\partial x_{1 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{2 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{3} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{4 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{5 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{6} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{7} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{8 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{9 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{10 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{11} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{12 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{13 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{14 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{15} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{16 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{17 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{18 } } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{19} } = ?[/tex]

[tex]\frac{\partial T}{\partial x_{20 } } = ?[/tex]

[Guest]

Remark: Quite interesting stuff, the partial derivatives of T! We will study them carefully. And hopefully, the information will help us to solve our problem.

[Guest]

Remark: The nature of T and X has changed according to our purposes. We apologize for any confusion.

[Guest]

FYI: 'Partial Derivative',

https://en.wikipedia.org/wiki/Partial_derivative.

Are we missing the point, X, for T(X) = k?

Time will tell.

[Guest]

FYI: 'Partial Derivative',

https://en.wikipedia.org/wiki/Partial_derivative.

Are we missing the point, X, for T(X) = k?

Time will tell.

"Hmm. Point us (mathematicians) in the right direction, and we shall arrive at a solution."

Some Ideas: Initial Values, ([tex]X_{0 }[/tex], T([tex]X_{0 }[/tex])[tex]= k_{0 }[/tex] ), Final Value (Solution), ([tex]X_{m }[/tex], T([tex]X_{m }[/tex]) [tex]= k_{m }=k[/tex] ), Gradient and Paths between points ([tex]X_{j }[/tex])/levels ([tex]k_{j}[/tex]), Neighborhood of points/levels, Convergence, and Uniqueness of Solution.

[Guest]

Remark: The existence and uniqueness of solutions are important!

[Guest]

Remark: Newton's Method (Jacobian) is quite useful here.

[Guest]

Remark: The existence and uniqueness of solutions are important!

There may be solutions galore, [tex]X \in \mathbb{R^{n}}[/tex], such that T(X) = k. But there could also be no solutions, [tex]X \in \mathbb{Z^{n}}[/tex], such that T(X) = k.

Go figure!

[Guest]

Remark: The existence and uniqueness of solutions are important!

There may be solutions galore, [tex]X \in \mathbb{R^{n}}[/tex], such that T(X) = k. But there could also be no solutions, [tex]X \in \mathbb{Z^{n}}[/tex], such that T(X) = k.

Go figure!

Exhaustive Search:

We assume T(X) = k and [tex]X \notin \mathbb{Z^{n}}[/tex].

Let [tex]X + \triangle X \ne X[/tex] such that T([tex]X + \triangle X[/tex]) = k.

Does the sum, [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex]?

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

[Guest]

Remark: The existence and uniqueness of solutions are important!

There may be solutions galore, [tex]X \in \mathbb{R^{n}}[/tex], such that T(X) = k. But there could also be no solutions, [tex]X \in \mathbb{Z^{n}}[/tex], such that T(X) = k.

Go figure!

Exhaustive Search:

We assume T(X) = k and [tex]X \notin \mathbb{Z^{n}}[/tex].

Let [tex]X + \triangle X \ne X[/tex] such that T([tex]X + \triangle X[/tex]) = k.

Does the sum, [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex]?

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

Remark: [tex]\triangle X \in \mathbb{R^{n}}[/tex] and [tex]\triangle X =[/tex] ([tex]\triangle x_{1 }[/tex], [tex]\triangle x_{2 }[/tex], ..., [tex]\triangle x_{n }[/tex] ).

[Guest]

Final Remark: David Hilbert's Tenth Problem is open!

Good luck!

Go Blue!