mabshoff wrote:
ed smaller values. I'm going to
>>> put that data up on the trac ticket.
>> Mathematica 6 (on a Sun SPARC) gives an answer in far less time than Sage:
>>
>> In[3]:= PrimePi[2^50]
>>
>> PrimePi::largp:
>> Argument 1125899906842624 in PrimePi[1125899906842624]
>> is too large for this implementation.
>>
>> Out[3]= PrimePi[1125899906842624]
>>
>> Well, perhaps not really an answer!
>
> :)
>
> Could you figure out what the upper bound is that MMA allows? I have
> discussed this with William in IRC and in 3.4.2 we should just throw a
> NotImplementedError for some bound where we are comfortable with
> knowing the result is correct on 32 and 64 bit. Unfortunately this
> isn't something we can doctest with a reasonable amount of time.
>
> The suggestion then was to implement something on top of the range
> computed with floats using MPFR for example, but we will see what
> happens. I am sure that if I asked if someone needed to compute
> prime_pi() for anything larger than 2^48 someone would say yes, so
> this ought to be fixed.
>
> Cheers,
>
> Michael
Hi Micheal,
Yes, I can figure it out.
The upper limit is PrimePi[249999999999999] (2.5x10^14), which
Mathematica gives as 7783516108362. It took 20 minutes or so on a
heavily loaded machine.
In[95]:= PrimePi[249999999999999]
Out[95]= 7783516108362
It can not manage PrimePi[250000000000000]
2^47 is 1.41x10^14,
2^48 is 2.81x10^14.
Since the maximum that can be handled is just under 2.5x10^14,
Mathematica can compute PrimePi[2^47], but not PrimePi[2^48]
Here's a table of PrimePi[2^n], with n ranging from 0 to 47. It took
roughly 20 minutes or so to compute the table.
In[19]:= Table[{n,PrimePi[2^n]},{n,0,47}]
Out[19]= {{0, 0}, {1, 1}, {2, 2}, {3, 4}, {4, 6}, {5, 11}, {6, 18}, {7, 31},
> {8, 54}, {9, 97}, {10, 172}, {11, 309}, {12, 564}, {13, 1028},
> {14, 1900}, {15, 3512}, {16, 6542}, {17, 12251}, {18, 23000},
> {19, 43390}, {20, 82025}, {21, 155611}, {22, 295947}, {23, 564163},
> {24, 1077871}, {25, 2063689}, {26, 3957809}, {27, 7603553},
> {28, 14630843}, {29, 28192750}, {30, 54400028}, {31, 105097565},
> {32, 203280221}, {33, 393615806}, {34, 762939111}, {35, 1480206279},
> {36, 2874398515}, {37, 5586502348}, {38, 10866266172},
> {39, 21151907950}, {40, 41203088796}, {41, 80316571436},
> {42, 156661034233}, {43, 305761713237}, {44, 597116381732},
> {45, 1166746786182}, {46, 2280998753949}, {47, 4461632979717}}
PS, Mathematica computes PrimePi[some_negative_number] as 0. Does Sage
handle that case ok?
Dave
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