Eighth Order PHI – Fermat's Third Prime: 17
Fermat devised a series of integers based on this formula: 22n + 1, wherein n = {0, 1, 2, 3, 4}. We'll ignore his premise of these being an infinite series of primes, because I'm only interested in their associated series of numbers: 2(2n − 1), wherein n = {0, 1, 2, 3, 4, ...?}. And for the purposes of this webpage, I'll only focus on the third 'n' of n = 2, thereby: 2(22 − 1) = 2(4 − 1) = 23 = 8. The eighth order Golden Series of Numbers and Golden Integers and its associated Golden Polynomial are all embedded within Fermat's third prime for this series, namely: the 17-sided regular polygon, or heptadecagon...
PHI series of numbers, 8th order:
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echo '' . "\n";
echo 'a' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'b' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'c' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'd' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'e' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'f' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'g' . "\n";
echo '' . "\n";
echo ' | ' . "\n";
echo '' . "\n";
echo 'h' . "\n";
echo '' . "\n";
echo ' |
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' . "\n";
echo '| ' . "\n";
echo $diagonals[0][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[1][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[2][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[3][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[4][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[5][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[6][0] . "\n";
echo ' | ' . "\n";
echo $diagonals[7][0] . "\n";
echo ' |
' . "\n";
for ($i = 1; $i <= $print_limit; $i++)
{
echo '| ' . "\n";
echo $diagonals[0][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[1][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[2][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[3][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[4][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[5][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[6][$i] . "\n";
echo ' | ' . "\n";
echo $diagonals[7][$i] . "\n";
echo ' |
' . "\n";
}
?>
' . "\n\n";
echo 'After ' . $print_limit . ' iterations, the approximation of the eight roots of the 8th order of PHI accurate to eleven decimal places are... ' . "\n" . '
' . "\n";
echo 'X1 = a/h = ' . $diagonals[0][$print_limit] . ' ÷ ' . $diagonals[7][$print_limit] . ' = ' . $root1 . '' . "\n" . '
' . "\n";
echo 'X2 = b/f = ' . $diagonals[1][$print_limit] . ' ÷ ' . $diagonals[5][$print_limit] . ' = ' . $root2 . '' . "\n" . '
' . "\n";
echo 'X3 = c/d = ' . $diagonals[2][$print_limit] . ' ÷ ' . $diagonals[3][$print_limit] . ' = ' . $root3 . '' . "\n" . '
' . "\n";
echo 'X4 = d/b = ' . $diagonals[3][$print_limit] . ' ÷ ' . $diagonals[1][$print_limit] . ' = ' . $root4 . '' . "\n" . '
' . "\n";
echo 'X5 = e/a = ' . $diagonals[4][$print_limit] . ' ÷ ' . $diagonals[0][$print_limit] . ' = ' . $root5 . '' . "\n" . '
' . "\n";
echo 'X6 = f/c = ' . $diagonals[5][$print_limit] . ' ÷ ' . $diagonals[2][$print_limit] . ' = ' . $root6 . '' . "\n" . '
' . "\n";
echo 'X7 = g/e = ' . $diagonals[6][$print_limit] . ' ÷ ' . $diagonals[4][$print_limit] . ' = ' . $root7 . '' . "\n" . '
' . "\n";
echo 'X8 = h/g = ' . $diagonals[7][$print_limit] . ' ÷ ' . $diagonals[6][$print_limit] . ' = ' . $root8 . '' . "\n" . '
' . "\n";
?>
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