Will,

Could you please move 3 posts above to some forum, "How do I...?" They have nothing to do with the topic of this blog.

The question, as I understand it, is about RSA codes - given a 400-digit number that is a product of 2 primes, find these primes.

The given above answer is completely unrelated to that - it says that to find the product of 2 numbers, you have to multiply them.

Then, it uses nextprime command that gives not primes, but only pseudoprimes, as all of us know.

Third, the suggested pseudoprimes, even if they were primes, are not usable for the RSA code - it is not the worst possible case, the worst possible case is when one of primes is 5 - then, if the second prime is greater than 2, the product has last digit 5; and it is not the second worst posiible case - the second worst possible case is when one of primes is 2 - then the product is even. I think, it shares the third worst case with the case when one prime number is 3. It can be easily cracked not only by taking nextprime and prevprime to the square root of the product, but in many other ways - such as adding squares, 1^2, 2^2, 3^2, etc. to the product and checking whether the result is a square - then if n+i^2=a^2, then n=(a-i)*(a+i). Not talking about complexity - both suggested (pseudo)primes and their product consist mostly of 0s.

Alec

As far as I can tell, the right to use the common law trademark, i.e. the TM sign, belongs to the founders of Maple project.

I could pretend, with the amount of time that I invested in Maple, on some moral rights though - but these moral rights haven't warrantied me even a complimentary copy of Maple 12, not talking about mugs, bags, pens, and T-shirts.

Alec

That was very useful!

Thank you very much!

Alec

Not surprising that Maple is not #1 (and now, with SAGE in play, probably, not #2 either.)

Alec

Here.

Alec

Here is what William Stein, the leader of SAGE project (who started it few years ago in Harvard), said about Calculus in SAGE,

"Sage does not use Axiom for anything.   Sage uses Maxima as
a backend for a lot of Calculus right now.  In the long run it
will likely use Maxima less and new faster more modern code
that it is in the pipeline (this is work being funded by Google,
but not as part of Google Summer of Code).    It will be interesting
to see how all of these physics/mathematicians/etc. assumptions
will play out in Sage as compared to how they played out in
Maple.

It is likely in Sage that we'll have for calculus
a global proof=True and proof=False
mode, like we have with number fields, linear algebra, etc.
With proof=False, assumptions about partial commuting,
functions being continuous, etc., like maple makes, would
be in force.  The default unless explicitly changed would be
proof=True.  One could see everywhere in the source where
the proof flag is used, hence see precisely what assumptions
are being made in a computation..."

Alec

 

Not related to the correctness of the calculations, could somebody who has Sage, Mathematica, Maple, and Pari on the same computer, post the timings of calculations of the number of partitions function for, say, powers of 10, such as 10^4, 10^5, 10^6, 10^7, 10^8, and 10^9 in these systems?

I have everything else except Maple, so that wouldn't be especially interesting for Mapleprimes.

Alec

That, probably, was the first time that I got sorry that I don't have Maple installed. Is there a Maple Player for viewing .mws files, available for free downloading (as Mathematica has for viewing .nb files)?

Alec

Just found out out that searching Google for Clairaut's theorem gives this thread as the third link on the page (after Wikipedia and PlanetMath.)

Alec

Yes, that's it.

It was $100 money but much more prestige. I think, Carl DeVore's team also won the first prize.

Alec

One thing that I can say about Gaston Gonnet, is that he is a very good mathematician. Once I posted in sci.math.symbolic a reply to another post including an asymptotic of some values of Gamma-function using Laplace method and Maple. And almost immediately he replied (in another forum) with much more generalized statement (also including Laplace method and Maple.) That was very impressive!

Also, if I am not mistaken, the team of him and Robert Israel won the first prize ($100) in some symbolic and numerical math competition that I forgot the name of, few years ago. 

Alec

The problem with current definition of the functions for wich diff and D can be applied,

The diff command assumes that partial derivatives commute

is that unless the function is known to be analytical, there seem to be no other way to tell whether the derivatives commute, or not, as to calculate them. And calculations using diff and D could be done only if you know that the derivatives commute. It is a vicious circle.

Perhaps it would be better to say that diff and D assume that functions are analytical?

Alec

Alejandro,

Thank you very much for the link. Both the blog and the links in it are very interesting.

Alec

Certainly true. And textbook prices are ridiculuous. Perhaps, I should say: "it is good to be rich, live in the US etc." But then, I could stop earlier, saying just "It is good to be rich" :)

Alec

PS and "healthy" -Alec

Just tried it in SAGE. The situation seems to be similar to Maple. For symbolic variables it gives

sage: x,y=var('x,y')
sage: f=x*y*(x^2-y^2)/(x^2+y^2)
sage: f.diff(x)(0,y).diff(y)
-1
sage: f.diff(y)(x,0).diff(x)
1

that is OK, but trying to evaluate second derivatives at 0 gives division by 0 error,

sage: f.diff(x,y)(0.,0.)
NaN
sage: f.diff(y,x)(0.,0.)
NaN

and it also gives

sage: f.diff(x,y)-f.diff(y,x)
0

That actually is also OK in this situation though, as well as in Maple, because the function f is not defined at 0. That would be wrong if the function was continued as in the Wikipedia to being equal 0 at 0. I'll try to figure out how to do that.

I didn't install Mathematica yet. Couldn't decide where to install it - in Linux, or in Windows (I have only one license). But it is interesting to know whether it can do that, or not.

Alec

PS One might want to have 1/x-1/x, for example, being undefined at 0 and equal 0 otherwise, instead of being equal 0 - but that may be too much to ask. -Alec