I've noticed two issues with random_prime() in Sage.
1) Whilst Sage's random_prime() looks as though it will work for numbers to at
least 10^100000000, if one specifies a lower bound, then the upper bound reduces
to 2^40, which is about 10^12.
This situation does *not* change if one specifies one only wants a pseudo prime.
sage: random_prime(10^13,proof=False, lbound=12)
<snip>
NotImplementedError: computation of prime_pi() greater 2**40 not implemented
http://trac.sagemath.org/sage_trac/ticket/10277
Mathematica 7.0.1 has no such limitation. One can set a lower bound, and still
have an upper bound to at least 10^100000000
2) The Sage implementation is incredibly slow compared to Mathematica.
On my machine (a 3.33 GHz Sun with !OpenSolaris) using a 32-bit build of Sage,
it took 1455 seconds to generate a random prime up to 10^1000.
Mathematica 7.01 does this in 7 seconds.
http://trac.sagemath.org/sage_trac/ticket/10278
If Mathematica does use a pseudo prime test, it is not documented.
http://reference.wolfram.com/mathematica/ref/RandomPrime.html
Dave
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