# On the Solution of Type One Diophantine Equations

### On the Solution of Type One Diophantine Equations

'Entia non multiplicanda praeter necessitatem (Entities should not be multiplied beyond necessity)." -- Occam.

We are given the Type One Diophantine Equation,

1. $$AX^{M} = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k$$,

and we ask the following questions:

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Question: What is a solution of equation one?

Question: How many solutions does equation one have?
...

Dave
https://www.researchgate.net/profile/David_Cole29
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:'Entia non multiplicanda praeter necessitatem (Entities should not be multiplied beyond necessity)." -- Occam.

We are given the Type One Diophantine Equation,

1. $$AX = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k$$,

and we ask the following questions:

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Question: What is a solution of equation one?

Question: How many solutions does equation one have?
...

Dave
https://www.researchgate.net/profile/David_Cole29

Notes:

Update: 1. $$AX = \sum_{i=1}^{l }A_{i }X_{i }^{M} = k$$.

A is a diagonal matrix such A = {$$a_{ij }= 0$$ if $$i \ne j$$, otherwise $$a_{ij }$$ is a nonzero integer}.

X is a column vector such that X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\$${0}^{l}$$.

$$X_{i }^{M}$$ = {$$x_{i }^{m}, x_{i}^{m-1}, x_{i }^{m-2}, ..., x_{i}$$} $$\in \mathbb{Z}^{l}$$\$${0}^{l}$$ where $$1 \le i \le l$$.
Guest

### Re: On the Solution of Type One Diophantine Equations

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...
Guest

### Re: On the Solution of Type One Diophantine Equations

\ge
Guest wrote:Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We formulate the following equation,

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} \ge 1$$,

over the set on nonzero integers such that equation one has at least one nonzero integer solution.
...

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:\ge
Guest wrote:Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = k$$ (Updated).

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$.

over the set on nonzero integers such that equation one has at least one nonzero integer solution.

3. $$N_{k+1} = N_{k} +J_{k}^{-1} * N_{k}$$ (Newton's Method).

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

We also think some knowledge/tools on algebraic curves will help us solve our problem...

Relevant Reference Link:

https://en.wikipedia.org/wiki/Algebraic_curve
Guest

### Re: On the Solution of Type One Diophantine Equations

Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

viewtopic.php?f=63&t=1793&start=60
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:
Guest wrote:\ge
Guest wrote:Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = k$$ (Updated).

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$.

over the set on nonzero integers such that equation one has at least one nonzero integer solution.

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

Relevant Reference Links:

https://www.math10.com/forum/viewtopic.php?f=63&t=8285

viewtopic.php?f=63&t=7803

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:
Guest wrote:Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793

FYI: 'Solving Diophantine Equations' by Profs. Octavian Cira and Florentin Smarandache:

http://fs.unm.edu/SolvingDiophantineEquations.pdf
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:
Guest wrote:
Guest wrote:Question: What does the truth of the ABC conjecture say about the solutions of Type I Diophantine equations?

Relevant Reference Link:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793

FYI: 'Solving Diophantine Equations' by Profs. Octavian Cira and Florentin Smarandache:

http://fs.unm.edu/SolvingDiophantineEquations.pdf

FYI:

'An Introduction to Diophantine Equations: A Problem-Based Approach' by Prof. T. Andreescu et al.
Guest

### Re: On the Solution of Type One Diophantine Equations

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...[/quote][/quote]

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = \sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M} + \sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M} + ...$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k$$ Updated);

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

Relevant Reference Links:

https://www.math10.com/forum/viewtopic.php?f=63&t=8285

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

Dave.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = \sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M} + \sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M} + ...$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k$$ Updated);

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

...

Dave.

Hmm. Because of equation 2 and because our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf
Guest

### Re: On the Solution of Type One Diophantine Equations

An Update:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||\sum_{i=1}^{j_{1 }}A_{1}X_{1 }^{M}|| + ||\sum_{i=1}^{j_{2 } }A_{2}X_{2 }^{M}|| + ...$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1j_{1}j_{1}}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2j_{2}j_{2}}x_{2} + ... = k$$.

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:An Update:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k$$.

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.
Guest

### Re: On the Solution of Type One Diophantine Equations

Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k$$.

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

...

Hmm. Because of equation 2 and because our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf
Guest

### Re: On the Solution of Type One Diophantine Equations

Since my work in mathematics has brought me no happiness nor awards, I have decided to retire from it today…

Goodbye and good luck to all,

David Cole,

https://www.researchgate.net/profile/David_Cole29
Guest

### Re: On the Solution of Type One Diophantine Equations

Guest wrote:Question: What conditions on equation one guarantee at least one solution, X, over the set of nonzero integers?

Hmm. The solution, X, must be a nonzero integer vector. Moreover, the sum of the distances (positive) between two distinct elements of X constrains equation one and can guarantee with conditions that the solution, X, is a nonzero integer vector.

...

We think the solutions, X = {$$x_{1 }, x_{2}, x_{3 }, ..., x_{l}$$} $$\in \mathbb{Z}^{l}$$\{0}$$^{l}$$, for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:

1. $$||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = ||A_{1}X_{1 }^{M}|| + ||A_{2}X_{2 }^{M}|| + ... + ||A_{l}X_{l }^{M}||$$
$$= a_{111 }x_{1 }^{m} + a_{122 }x_{1 }^{m-1} + ... + a_{1mm}x_{1 } + a_{211 }x_{2 }^{m} + a_{222 }x_{2 }^{m-1} + ... + a_{2mm}x_{2} + ... + a_{l11 }x_{l }^{m} + a_{l22 }x_{l }^{m-1} + ... + a_{lmm}x_{l} = k$$.

Note: Some but not all integer coefficients, $$a_{inn }$$, associated with each variable, $$x_{i }$$, could be zero.

2. $$\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1$$

over the set on nonzero integers such that equation one has at least one nonzero integer solution;

3. $$N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}$$ (Newton's Method).

...

Hmm. Because of equation 2 and because of our need for intelligent (efficient and directed) searches for solutions to equation one, we recommend some study of graph theory too.

Relevant Reference Link:

'Graph Theory' by Prof. Keijo Ruohonen,

http://math.tut.fi/~ruohonen/GT_English.pdf

Relevant Reference Link:

'THE INSOLUBILITY OF CLASSES OF DIOPHANTINE EQUATIONS' By Profs., N. C. Ankeny and P. Erdös,

https://pdfs.semanticscholar.org/cee7/94fa9154cd0b4ac9d3c453f85918bf732bb1.pdf.

Remark: This paper is quite insightful! Enjoy!
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### Re: On the Solution of Type One Diophantine Equations

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