Recent content by Xevarion

There are no cubes which are 4 apart. The difference between consecutive cubes grows rapidly (quadratically).

Just so you know, this is a hard problem in general... http://en.wikipedia.org/wiki/Elliptic_curve#Connections_to_number_theory

I'm pretty sure it's 0. See http://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number

I think the OP wants to know if there's a known closed form for \Gamma(i), meaning (presumably) a finite combination of elementary functions and algebraic numbers. (So no infinite products, no integrals, and no "So-and-so's constant".) The answer: I have no idea. My guess is that if there even...

"Continuously differentiable" is not the same as "complex differentiable," as far as I know. Make sure you've checked how the book defines those terms, and what sort of function you're looking at. If a function is complex analytic, then it is smooth (ie ANY derivative is continuous, not just...

they are all isolated points.

That story is supposedly about Hardy.

I think the fact that this conjecture is false would follow from the ABC conjecture. I didn't work it out carefully but it seems reasonable. (At least the previous one, with 2 and 3 only, would follow from ABC. And this problem seems to be essentially the same.)

Yeah. Luckily if your x_i are far apart in (b), you can definitely replace that sum with the biggest x_i (I mean least negative) times a small constant (like 3/2 or 2 maybe, depends on how much smaller the other terms are).

Reasons to expect the conjecture to be true: If you recall, the earlier conjecture about powers of 2 and 3 was false because if you looked mod certain big numbers, the powers of 2 and 3 didn't hit that many residues, so differences of them didn't hit all residues. But since we now get to use all...

OK I think we can show the conjecture is true for sufficiently big k by the following ideas: (1) There exists a primitive root mod n which is smaller than cn^{1/4} for some absolute constant c when n is sufficiently large. (2) Observe that we can make any number between 2 and p_k >...

That sum doesn't converge. There are arbitrarily large $m, n$ with $|\sin(n)\sin(m)|$ close to 1.

Here's a simpler and more formal statement of your conjecture. approx's Prime Difference Conjecture Suppose that p_k is the k^{th} prime, and k \ge 3. Then for each prime q with p_{k-1}^2 < q < p_k^2 there exist positive integers m, n such that (1) q = m - n (2) If j < k then p_j \mid m or...

You should never assume \|x\| is the L^2 norm, unless the author has stated that explicitly. It just means 'the norm of x'. Which norm should be clear from the context; if not, go back a few pages and try to figure it out. I don't think I have ever seen |x| used for any norm other than the...

Either way the limsup is 1. Who cares if there are a lot of 0s? There are also a lot of times when it will be very close to 1. A more interesting question is whether the liminf is 0 if n is an integer... the hint would be to use a Diophantine approximation result to see how close |sin(n)| gets...