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16

If you calculate Log[2,Log[2,$MaxNumber]], you'll get 29.999999828017338886225739 which is remarkably close to 30. Therefore I conclude that Mathematica calculates with a 31-bit exponent (1 bit for the exponent's sign). Which means that if Mathematica uses the same ordering as IEEE floats (i.e. first sign bit, then exponent, then mantissa), the first 32 ...


16

It's due to an implementation-dependent issue. We should try to improve on it. Has not been much clamor to do so, therefore it has not been a high priority. --- edit --- I've had a look at the code. It is quite intentional that the largest is around what you state (I see the constant being set to $7.783516108362\times 10^{12}$). It has to do with this ...


13

Actually, I believe the issue reduced to that of implementing PrimePi[]. It is easy to implement Prime[] using PrimePi[] and FindRoot[] — in fact this is done on page 134 of Bressoud and Wagon, "A Course in Computational Number Theory". So all you need is to have a fast implementation of PrimePi[]. The first efficient way was found by Legendre in 1808. The ...


8

As it seems to depend on more than machine bits I'm curious what $MaxNumber various Mathematica installs have. If your setup is different please fill in system information and Log2 @ Log2 @ $MaxNumber // Round in the table below: $$\begin{array}{r|c|c|l|c} \text{OS} & \text{Bits} & \text{Version} & \text{\$MaxNumber} & \log_2\log_2\\ ...


5

The answer was already given by Xerxes. It is a limitation of MIDI, which only supports 16 channels. That Mathematica really uses MIDI is described in the docs. For instance in the tutorial The Representation of Sound: At the lowest level, all sounds in Mathematica are represented as a sequence of amplitude samples, or as a sequence of MIDI events.



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