Guest wrote:'Entia non multiplicanda praeter necessitatem (Entities should not be multiplied beyond necessity)." -- Occam.
We are given the Type One Diophantine Equation,
1. [tex]AX = \sum_{i=1}^{l }A_{i }X_{i }^{M_{i }} = k[/tex],
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 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.
...
Guest wrote:\geGuest 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.
...
Guest wrote:Guest wrote:\geGuest 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 = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:
1. [tex]||\sum_{i=1}^{l\ge 2 }A_{i }X_{i }^{M}|| = k[/tex] (Updated).
2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex].
over the set on nonzero integers such that equation one has at least one nonzero integer solution.
3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (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 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 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
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
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 = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:
1. [tex]||\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} + ...[/tex]
[tex]= 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[/tex] Updated);
2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]
over the set on nonzero integers such that equation one has at least one nonzero integer solution;
3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (Newton's Method).
...
Dave.
Guest wrote:An Update:
1. [tex]||\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}||[/tex]
[tex]= 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[/tex].
Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.
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 = {[tex]x_{1 }, x_{2}, x_{3 }, ..., x_{l}[/tex]} [tex]\in \mathbb{Z}^{l}[/tex]\{0}[tex]^{l}[/tex], for Type I Diophantine equations require some optimization/convergence of relevant parameters/variables and some clever algebraic manipulation/programming involving the following equations:
1. [tex]||\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}||[/tex]
[tex]= 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[/tex].
Note: Some but not all integer coefficients, [tex]a_{inn }[/tex], associated with each variable, [tex]x_{i }[/tex], could be zero.
2. [tex]\sum_{i\neq \forall j\geq 1}^{l \ge 2}|x_{i} - x_{j}| = k_{perimeter} = k_{p} \ge 1[/tex]
over the set on nonzero integers such that equation one has at least one nonzero integer solution;
3. [tex]N_{i+1} = N_{i} +J_{i}^{-1} * N_{i}[/tex] (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
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