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Immortality of Quaternions

Approximation for various powers at base 10, using immortal Quaternions with 1-$max_digits digits and only positive parts;
\n"; echo "2 is subtracted from the amount of integers in order to exclude the super immortal numbers 0 and 1:

\n\n"; echo "

[latex]\\vert\\mathbb{M}^{power}_{10}\\vert^{".$symbol."^{+}} \\approx \\tfrac{true\\ quaternion\\ numbers + pure\\ complex\\ numbers + integers - 2}{{".$max_digits."}^{".$dim."}}[/latex]

\n\n"; for ($power=2; $power<=$max_power; $power++) { $out = []; exec('zcat result_'.$max_digits.'digits_maxpower'.$max_power.'.txt.gz | grep " ^ '.$power.' " | grep "REAL" | wc', $out, $wc); $real = explode(' ', trim($out[0]), 2)[0]; $out = []; exec('zcat result_'.$max_digits.'digits_maxpower'.$max_power.'.txt.gz | grep " ^ '.$power.' " | grep "CPLX" | wc', $out, $wc); $cplx = explode(' ', trim($out[0]), 2)[0]; $out = []; exec('zcat result_'.$max_digits.'digits_maxpower'.$max_power.'.txt.gz | grep " ^ '.$power.' " | grep "QUAD" | wc', $out, $wc); $quad = explode(' ', trim($out[0]), 2)[0]; echo "

[latex]\\vert\\mathbb{M}^{".$power."}_{10}\\vert^{".$symbol."^{+}} \\approx \\tfrac{".$quad." + ".$cplx." + ".$real." - 2}{{".$max_digits."}^{".$dim."}} = ".(($quad+$cplx+$real-2)/(pow($max_digits,$dim)))."[/latex]

\n\n"; } ?>