@Mac Dude You misunderstand my point entirely. I was commenting on your continuing the OP's use of f^4; I was not commenting on your use of f@@4 without arguments. Indeed, my Reply was added before you had edited your Reply to use @@ at all. In other words, I was the first person in this thread to make any reference to the @@  operator, and I did so merely to make readers aware of its existence as the proper Maple operator to use for what was up until my intervention being refered to as f^4.

I have no problem with using f@@4 without arguments to refer to the fourth-order self composition. And if you do include arguments, it must be (f@@4)(x,y), not f@@4(x,y).

Please post an example of the phenomenon that you describe, either as plaintext or a Maple worksheet.

@Norwegian explorer You have two procedures named i. So, the first one is never used and never could be used. 

Also, the second procedure i uses i as a local variable. This is not a problem, but it's a bit unusual.

Your weight function omega is not strictly real-valued for x in [0,1]. Isn't that a problem?

I think that this one takes the cake as the most irrelevant Maple error message that I've ever seen. 

@Mac Dude To avoid abusing the notation, the Maple syntax for the object under discussion is (f@@4)(x,y).

@syhue It is not biased.

The thing that I was saying is biased is

nextprime(rand(...)())

@Rouben Rostamian Thanks, I had just come to the point of guessing that.

@vv Perhaps I don't understand what "concurrent" means. I'd guess that it means that They are along the same line. But I can only imagine pairs, not quadruples, of points on an ellipse having that property, and they'd be diametrically opposed. Does it instead mean that they intersect at a common point?

Considering the specific operation that you ask about timing---modular exponentiation---and that you emphasized a desire for its exact or precise time, I wonder if your interest is exploring the possibility of a so-called side-channel attack on RSA? Specifically, depending on the algorithm used for modular exponentiation, with some experimental timing data and a knowledge of the precise timing of C&^d mod n, one can figure out the total number of bits and the number of one bits of the decryption key d. One may be able to figure out something else about d also----of that I'm not sure. 

@ntan113 I still see no attachment. What is the name of the file that you're trying to attach? Sometimes weird names are the problem. Please also state what Maple version you're using.

@syhue 

Tom is correct that &^ (like many functions) is only defined when it's used in conjunction with mod. Anyway, there'd be no point in using it for a trivial one-word computation like 2^50.

Regarding selecting a prime factor at random: I assume that your w will always be even and that you never want to select 2. In that case, do

combinat:-randcomb(ifactors(w)[2][2..], 1)[][1];

Regarding selecting a prime at random from an interval: For better crypto-security, you probably want to select it uniformly at random. The technique that you used is commonly used, but it is biased towards primes that are at larger distances from their predecessors. To make a uniform (i.e., unbiased) random selection, use

UniformRandPrime:= proc(R::range(And(realcons, positive)))
local RI:= rand(ceil(lhs(R)) .. floor(rhs(R)));
   proc() local r; do r:= RI() until isprime(r) end proc
end proc
:
URP:= UniformRandPrime(2^50..2^51):
(p,q):= 'URP()'$2;

I assume that you meant to attach some code or a Maple worksheet. Nothing got attached. Please edit your Question and try that again. 

Is your actual problem that you want to find the value of the model parameter a such that the level surface UC(...) = 1 best approximates the experimental data?

A Google search for "Joaminsthal" returns 0 hits. So, this name doesn't appear in a single web page anywhere in the world?

I assume that your Question is essentially Does this Maple code and its accompanying comments fulfill the requirements of this assignment, i.e., does it verify the theorem? Without a clear reference for the theorem, I can't Answer that.