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.

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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).

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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