All differential equations have dependent functions.

@Rouben Rostamian  I took it to be not an equation---a statement of equality---but that the left side (in particular, a) was being (incorrectly) defined to be the right side.

Some of your "points" in L have more than two coordinates. What do you mean by that?

You're confusing a plot, which is 2D, with a plot3d. For a plot of a curve, the second column of the data matrix contains the y-values (the vertical coordinates). That's not true for a plot3d. For the most-basic form of plot3d, the GRID form (which is what you have), the entire matrix is z-values. Preben's Answer uses this fact.

@CGForbes You have three curves, although they are given by only two formulas, so you need three matrices. Change (M1,M2) to (M1,M2,M3).

@CGForbes Please show me the result of just plottools:-getdata(P). Perhaps you have a version of Maple with a different version this command. If so, it'll still be easy to extract the data matrices. So, please also show me the result of indets(P, Matrix).

I think that the best treatment for such an expression (an algebraic number in radical form) is evala(S). The mathematical nature of the set of expressions that can be output by simplify (without optional arguments) isn't clearly documented. I wouldn't be surprised if that set wasn't even fully understood by Maple developers.

If Acer's Answer solves your problem, great. If not, tell me the size of your worksheet (the saved .mw file) in bytes.

@pake Adding 100,000 integers is still too small a task. Use 10 million integers. Also, the number of tasks that can actually be run in parallel (that is, running at exactly the same time) is processor dependent and given by kernelopts(numcpus). It's likely to be 1, 2, 4, or 8. It's not a problem if you ask for more tasks in parallel; the operating system handles the scheduling. But this will limit your speedup-factor expectation. If your numcpus is 1, there's no practical point in trying parallel processing, although measuring the slowdown factor that it causes is still interesting.

Since you think that the task size is irrelevant, I suggest that you measure the overhead by making your addproc a procedure that does nothing:

addproc:= proc() end proc;

@pake The computer's processor (or chip) model is more important than the operating system, in particular the number of "cores". Use the Maple command kernelopts(numcpus) to see how many simultaneous threads can run on each of your computers.

Also, as pointed out in the Answer below, your basic task of adding the integers from 1 to 1000 is much too small for this test. Change that to about 10,000,000 and reduce the number of iterations of the outer loops.

@vv Your solution leads to the very unusual situation that the definition of a module procedure depends on a global variable and if that global variable changes---even after the module is loaded---then the module export will change. It's not a bug per se, but I think that it's bug prone.

@vv But the divisors of a^b can be easily determined from the divisors of a.

@taro You should think of LL as a matrix which contains only the z-values. So, there are 12 z-values laid out in a 3 x 4 grid.

@Mitterrand Perhaps there is a bug with the limits in your version of Maple which has subsequently been corrected? If I re-execute your worksheet, I don't get 0. I get the same result as shown by Kitonum below.

@Kitonum Why gpwr:= gpwr*(g mod p) rather than the original gpwr:= gpwr*g mod p? I think that the original is correct.