Take the following Diophantine equation.
$$9923w -9187x +9011y +9973z = 0$$
Mathematica solves this as:
$$y = 9973 n + 4437 w + 6843 x, z = -9011 n - 4010 w - 6182 x, n \in \mathbb{Z}$$
How can you do this by hand?
2026-05-06 02:23:59.1778034239
Solving the Diophantine equation $9923w -9187x +9011y +9973z = 0$
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A reduced basis for your lattice is given by the rows of $$ \left( \begin{array}{rrrr} 3 & 17 & -7 & 19\\ 7 & -6 & -26 & 11\\ -19 & -11 & -8 & 16\\ \end{array} \right) $$
Given a triple of integer coefficients $a,b,c,$ the linear combination is the matrix product
$$ \left( \begin{array}{rrr} a & b & c \\ \end{array} \right) \left( \begin{array}{rrrr} 3 & 17 & -7 & 19\\ 7 & -6 & -26 & 11\\ -19 & -11 & -8 & 16\\ \end{array} \right) $$