physics

# Newtonism Question

A mass point of 1 unit is at 1 length unit.

A mass point of 1/2 unit is at 1/10 length unit.

A mass point of 1/4 unit is at 1/100 length unit.

….

Generally, it’s (1/2)^n mass at (1/10) ^n unit for every natural n, on the positive side of 0.

All gravitational net forces point to the left. Where is an opposite reaction force here?

Standard
mathematics

# Unexpected?

When you go up the natural numbers ladder, there are (on average) bigger and bigger gaps between consecutive prime numbers.

If you look at primes as islands, there are  lakes of composites between them.

The lake between 19 and 23 is 3 composite numbers long, for example. Or between 89 and 97 there are 7 composites.

Those lakes are greater and greater on average, as it is well known. Albeit there are always small ponds of just 5 composite numbers long, no matter how far you go. (There are likely infinity many lakes with just 1 composite, too. Which is another form of the so called Twin Primes Conjecture. A few years ago it has been proved that there are always less than 70000000 large lakes as you go up the naturals. Some time later, that 70 million has been improved to just 5, but not yet to 1.)

Say, that you have just sailed across some recordly wide composite lake and you are on a prime island again. What can you expect, how much wider will the next record lake be? Usually 2 more composites wider than the previous largest, right?

No. Every new record is usually quite greater from the previous one.

`1,2,2,2,6,4,2,2,12,2,8,8,20,14,10,16,2,4,14,16,6,26,30,10,2,12,14,2,32 ...`

These are the increments of every new record composite lake.

P.S. Haven’t been able to Google this story.

EDIT:

Not as smoothly and gradually as those three coloured functions predict, but quite abruptly in reality.

Standard
mathematics

# Set Theory Problem

There are at most aleph-zero disjunct 3D spheres in 3D space. And there are at least aleph-one disjunct 2D circles in every finite volume part of the 3D space?

The number of points in N dimensional space is always aleph-one. And you can also divide this space into aleph-one disjunct N-1 dimensional spheres.

Is there a way to divide 3D space into aleph-one cubes with no common volume? They may touch each other, but may not share some common volume.

If you can find one such space division, you brought down the ZF Set Theory.

EDIT:

Discussion there: