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I recently played around with wolframalpha, and I found a very interesting property.

This function: $$f(x)=\sin(\cos(x-1)!)$$

yields a negative output for most prime numbers when only integer inputs are graphed. The beginning of the function (close to $y$-axis) is very interesting. For $x=3, x=7, x=11, x=13, x=17, x=23, x=29, x=31, x=47$ which are all primes, the output is negative.

Of course, later on there are less prime inputs that give a negative output, but still prime number inputs dominate the function's negative output values.

Why does this happen, can this be somehow explained?

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There is no specific reason whatsoever in this observation. First of all, the graph of $cos((x-1)!)$ oscillates faster and faster as $x \to \infty$.Clearly, the sign of $cos((x-1)!)$ determines the sign of $f(x)$. If we considered reals, after moving along the X axis for a while, when we change from $x$ to $x+r$ [$r \in \mathbb R$], $cos((x-1)!$ may give drastically different results. That is, this function becomes more and more 'chaotic' as we move towards $+\infty$. During the initial region, the integers you checked for were lucky to give negative values. But as we moved away, the chances of finding such close values became random. You may or may not find a cluster of some 5-6 integers for which it is negative. This has got nothing to do with primes in particular. As such, analysing this as a chaotic system might be interesting.

here is a picture of cos(x-1)!

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  • $\begingroup$ Can I ask by the way what program is that? $\endgroup$ – KKZiomek Feb 14 '17 at 5:13
  • $\begingroup$ the software is geogebra $\endgroup$ – Lelouch Feb 14 '17 at 5:56

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