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It might be worth noting that this is a "pigeonhole principle" problem, but I'm not sure how to use PHP with it.

The mantissa of $\pi$ is the fractional part of it (i.e. everything after the decimal place). A question from class asks me to prove that there is a number within the infinite set $\{1\pi, 2\pi, 3\pi, 4\pi, ...\}$ such that taking its mantissa will give me a number less than $0.001$.

Using a computer, I can actually find out that $10^{600}\pi$ gives me a satisfactory result. That's... kind of cheating though. Approaching the problem in a more mathematical way, I've been looking for patterns in $\pi$ and trying to figure out if I could prove that as I increase to $n\pi$ where $n$ is some large number, the mantissa will be getting lower and lower.

So far, I have noticed that the first two digits of the mantissa of $8\pi$ are "$13$" while for $1\pi$, they are "$14$". This trend continues until $106\pi$, which gives me $0.008$. Aside from that, I haven't made much progress and am still iffy on how this relates to PHP.

Any thoughts?

M47145
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    Has nothing much to do with $\pi$, it is true for every positive irrational. The proof is by Pigeonhole argument. – André Nicolas Jan 19 '15 at 06:39
  • The proof is by a Pigeonhole argument. Has been done a number of times on MSE. Maybe Google Stack Exchange, irrational, Pigeonhole Principle, dense. – André Nicolas Jan 19 '15 at 06:45
  • You'll find the argument in this answer. – Brian M. Scott Jan 19 '15 at 06:51
  • How are you showing that with a computer that the fractional part of $10^{600}\pi$ is small? Seems like, unless you have a lot of digits of $\pi$, a computer program is going to be useless here. – Thomas Andrews Jan 19 '15 at 06:56
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    FWIW, there are much smaller values than $m=10^{600}: frac(m\pi) < 0.001$. In fact there are 99 such m under $10^5$, the smallest being 29486: $29486\pi \approx 92633.0009837$ – PM 2Ring Jan 19 '15 at 07:28

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