Listen to me, is there anything outside even though knowing everything is boring?
It doesn't matter if there's anything outside, because it don't fall into "EVERYTHING" column, so we just can't know what's outside the box, we can only make guesses without strong arguments, and this is not knowledge anymore.
hm, it makes sence - but I have other plans - imagine that everything around us is just not real, but it's some mental simulation of the real world, so that everybody is actually a simulation inside yourself. But then you and your mental imgination stimulate the science, and without knowing, YOU discover the scientifical results, because the people are controlled by YOU.
So here's the BIG BANG - you already KNOW EVERYTHING, but you just don't know that you know it
ohh, I understand now, you're talking about the exponents! But then these functions will no longer be polynomilas. I don't have any clue about the structure of such numbers, and it's likely that there's no theory of these at all, because before even thinking about the structure of the set of roots of such all such functions, you need some information about these roots - bounds, number (i suspect there may be infinite number of roots of such a "polynomial" with "finite" degree).
I don't think dealing in general with numbers like
hm interesting, but i'm stuck.
first of all, notice this (it may actually be the key to the problem if someone uses some strange trigonometric identities):
Actually, plying whit it a little, I noticed that z = -1/4 is a solution to
blah, surely the solution will be some involving tirgonommetric identities.
So, the algebraic number field is algebraically closed, this is a foundamental property of the field, and means exactly that every root of a polynomial with algebraic coefficients is algebraic too. This comes from basic factorisation properties of plynomials and using the fact that every first degree polynomial over the algebraic numbers has an algebraic solution.
OK, here's the whole thruth.
Take the the sums of squares. The sequence starts like this:
And we want to determine how is this sequence generated. There's a method, maybe it's called the difference method, which may help us, but ONLY if the generator of the sequence is a polynomial.
We put the numbers in rows like this:
0 1 5 14 30 1 4 9 16 3 5 7 2 2
where under two numbers is their difference. If the generator is polinomial, this process will eventually lead us to a row in which all the numbers are the same (or, which is the same, to a row in which all the numbers are 0)
Call the furst number on the first row, the first number on the second - , and so on till the first number of the nth row - . Then we use these coefficients in the "magic" formula:
In the case of sums of squares, we get:
First, the binomial coefficients have their combinatorial significance, so actually, what you're doing may seem only algebraical, but it's combinatorial. For small numbers (like 56) it's really most prefferable to use trial and error, because it will lead to solution faster. I can think of some kind of optimisation, where you don't actually try all the numbers. It may be used to tell if some number is binomial. There's a formula for the order (the maximal degree of a prime number p) which divides n!, that is:
You got me!
So... Since 1500 = 37.40 + 20, then
I think you've undestood, just one thing - we don't call the point a, every point on the curve is uniquely determined by its x-coordinate (there aren't 2 points with the same coordinate). And the derivative is not a distance, it's the slope of the line which approaches this so-called point a.
To get the last 2 digits you need the remainder modulo 100. First find the phi of 100:
And if you don't beleive, the number is actually
(Except in the boring case when A=0 and nothing happens ever)
No, you don't mean this - the zero exceptions give color to maths. A space becomes funny when we find some distinguishable elements of it. The zero case often is called trivial, but it's not trivial at all - it's foundamental.
Here's some nice way to visualise the process: you start with 2 sticks (lengths m,n), you put them one over the other with 1 common end, and you cut at the other end. Then remove one of the same sticks, which you just made, and continue. It's intuitial that you'll eventually end with 2 identical sticks, unless the initial lengths dont' have gcd (if you start with sticks with length 1 and pi for example, the process won't stop but will converge). This thing looks like the Ecld gcd. Formally, at each step the difference between the lengths of the sticks strictly decreases (which is not obvious using rationals, because they give us the relative difference between the two sticks - the numerator and denominator). The last thing is that just before the sticks become with same length, the one is twice the other.