I was playing around with $\tan e^n$ and realized that $\tan5^n$ has an interesting property:$$ \begin{eqnarray} \tan 5^{1}&=&-3.380515\ldots\\ \tan 5^{2}&=&-0.133526\ldots\\ \tan 5^{3}&=&-0.782060\ldots\\ \tan 5^{4}&=&-0.178807\ldots\\ \tan 5^{5}&=&-1.221283\ldots\\ \tan 5^{6}&=&-3.364104\ldots\\ \tan 5^{7}&=&-0.126782\ldots\\ \tan 5^{8}&=&-0.729953\ldots\\ \tan 5^{9}&=&-0.011144\ldots\\ \tan 5^{10}&=&-0.055778\ldots\\ \tan 5^{11}&=&-0.286042\ldots\\ \tan 5^{12}&=&-5.565505\ldots\\ \tan 5^{13}&=&-0.811792\ldots\\ \tan 5^{14}&=&-0.274453\ldots\\ \tan 5^{15}&=&-4.242139\ldots\\ \tan 5^{16}&=&-0.438533\ldots\\ \tan 5^{17}&=&1.8498218\ldots\\ \tan 5^{18}&=&-1.278866\ldots\\ \tan 5^{19}&=&-5.604437\ldots\\ \tan 5^{20}&=&-0.821873\ldots\\ \tan 5^{21}&=&-0.307247\ldots\\ \tan 5^{22}&=&-12.42162\ldots\\ \tan 5^{23}&=&-2.354334\ldots\\ \tan 5^{24}&=&0.4677197\ldots\\ \tan 5^{25}&=&-1.410674\ldots\\ \tan 5^{26}&=&17.137346\ldots\\ \tan 5^{27}&=&3.3336581\ldots\\ \tan 5^{28}&=&0.1141382\ldots\\ \tan 5^{29}&=&0.6384773\ldots\\ \tan 5^{30}&=&-0.309809\ldots\\ \end{eqnarray} $$It is worth noting that the first 16 $n$s and 22 $n$s out of the first 23 satisfy $\tan5^n<0$, in other words, $5^n\bmod\pi>\pi/2$.
If this is completely by chance, it seems like a little too much luck. I have an explanation for $n=9,10,11,12$: because $5^9\bmod\pi\approx0$. But the others seems to be a coincidence, and we can say that $5^9\bmod\pi\approx0$ itself is also a coincidence. Is there any explanation for this?
$\endgroup$ 3 Reset to default
