Math Is Fun Forum

It's very simple. And yes, it has connection to the d(x) and many more NT functions
I'll give you a hint: investigate the function:

Pictures 1-5:

I have played for a while with mathematica, and I founded interesting patterns.
The code:

ToCoords[x_][c_] := {Floor[c/x], c - x Floor[c/x]};
SqueezedDotPlot[x_, list_, ops___] := Show[Graphics[Point /@ (ToCoords[x] /@ list)], ops];
SqueezedLinePlot[x_, list_, ops___] := Show[Graphics[Line[ToCoords[x] /@ list]], ops];

Here I will actually use only the first plotting function.

Picture 1: this is the actual square function.

SqueezedDotPlot[10, Range[100]^2]

Picture 2: If you use irrational numbers, there can be interestiong results.

SqueezedDotPlot[4, Range[1000]^Sqrt[2]]

Picture 3: But for most of the numbers, you will get the ordinary noise.

SqueezedDotPlot[Sqrt[10], Range[10000]^3]

Picture 4: Some non-trivial structure

SqueezedDotPlot[Sqrt[2], Range[10000]^2]

Picture 5: The previous zoomed

SqueezedDotPlot[Sqrt[2], Range[1000]^2]

Picture 6: Noise again, but different from the ordinary.

SqueezedDotPlot[3, Range[10000]^1.5]

Picture 7: And what if the exponent is smaller than 1?

SqueezedDotPlot[2, Range[10000]^0.9]

Picture 8: interesting...

SqueezedDotPlot[1, Range[10000]^0.99]

Picture 9: What a wave!

SqueezedDotPlot[2, Range[10000]^0.999]

Picture 10: this is beautiful!!!

SqueezedDotPlot[0.1, Range[10000]^0.5]

Yes

Interesting. Here's a picture 10x10000:

Yes. You're right.
I like wikipedia, because i understand most of what i need.
someone may not.

Ricky wrote:

4|16 && 8|16 ,but 32 !| 16.
It's :
(a|c && b|c) =>gcd(a,b)|c.

For seven:
The topic is interesting. I think you're brave for posting it. I was not able to read your first posts, because they are DELETED!!!

You were asking for a statement, which cannot be proved with some set of axioms and which is true.
Here you're wrong. The "validality" of an statement depends on the set of axioms. There don't exist an universally true statemet, which is true for every set of axioms. So, if an statement A is unprovable over a set of axioms {X,Y,...,Z}, we can assume that it's true or it's false. If we assume that it's true, we are getting the new system: {X,Y,...,Z,A}, in which the statement is provable to be true.
But if we assume that A is false, over the set {X,Y,...,Z,!A}, A is provable to be false.

I hope you to understand.

The arcsin() function is commonly used for integration.

And one last note-strictly, fractional number of minutes won't be allowed, because we define the function (r) as a sum, and thus, it's only for integers (I'm not saying that it can't be defined over R, as we have done, I'm saying that in this case it should be defined only for integers (because we have drops, not stream))

Yes.
For example, 2^93=9903520314283042199192993792, which is very close to 10^...
But there's another problem-you can't practically use the above, because you must know fast way to evaluate 2^k for every positive integer k<93

You are right... - but the answer is a little bigger than 7 seconds, because in the 7th second, there will be 127 drops in the glass, but there must be 127.5, so half a drop must fall, to reach the half of the glass, so the time to full the half of the glass is bigger than 7 seconds with the time of falling of half a drop, which is really small.