I still had things to show from my math past

Part of this blog post is deleted... there were a lot of pictures here that are lost when republishing... however, here are two interesting statements worth sharing...

Statement 1:
We can make convex polyhedra from paper if we first construct their faces on paper, which meet at its vertices, and if we count those angles at these vertices that cover the areas which are NOT part of the surrounding faces (in theory), then we get 720 degrees.

Sketch of proof to Statement 1:
If F is the number of faces of the polyhedron, V is the number of vertices, and E is the number of edges, and N(x) is the number of vertices of face x, then we can say the following:

• The sum of inner angles of face x is (N(x)-2)·180 degrees, this is even taught in public education

• The sum of all inner angles of all faces is 180·((N(1)-2)+(N(2)-2)+(N(3)-2)+...+(N(F)-2))

• This is equal to 180·(2·E-2·F), or 360·(E-F)

• Checking out Euler's polyhedron theorem for convex polyhedra

• Euler's theorem can be used here, so this is equal to 360·(V-2)

• All the degrees at the vertices: 360·V

• So, all the degrees at the vertices minus the angles of the polyhedra at the vertices is 360·V-360·(V-2)=360·2=720 degrees, this is what our statement is about.

Edit: Since then, I learnt that this thing is already known, and called Polyhedral Curvature, and proven in a different way, which is also sketched on the AMS website.

Statement 2:
The statement of the (strong) Goldbach conjecture ("every even number greater than 2 can be written as the sum of two primes") is equivalent to the following statement: "For N > 1 natural number, every even number greater than N^2+1 can be written as the sum of two numbers that are greater than 1 and relatively prime to N!". See this link for more.

Sketch of proof to Statement 2:
Let P(i) be the "i"th prime number. If "Every even number greater than 1+P(i)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3), ..., P(i)", then let's write this condition for each of the primes: 

• Every even number greater than 1+P(1)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1)
• Every even number greater than 1+P(2)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2)
• Every even number greater than 1+P(3)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3)
• ...

• Every even number greater than 1+P(i)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3), ..., P(i) 

Of course, if the Goldbach conjecture is true, then all of these little statements are true, so our general statement is true as well, because primes are also relative primes at the same time. But does the converse hold? Or in other words, can these sums of numbers be sums of primes all the time, by choosing the right substatement?
For a number to be prime, it should not be divisible by any number less than its square root, except 1. So, if these two numbers are not divisible by any of P(1), P(2), P(3), ..., P(i), then consequently, their value is at least P(i+1), and if they are not prime (necessary for a counterexample), then their value is at least P(i+1)^2, altogether at least P(i+1)^2+P(i+1). If all the substatements in the above example are true until P(i), then the Goldbach conjecture is true from 1+P(i)^2+1 (and therefore, from 1+P(1)^2+1) until P(i+1)^2+P(i+1)-1 (at least, and below). Since we suppose all of them to be true towards 1+P(Infinity)^2, the Goldbach conjecture also comes from this sequence of substatements (i.e. true for even numbers 4-11, 11-29, 27-55, 51-etc.), which is otherwise expressed as a general statement we have proved now to be equivalent to the Goldbach conjecture.

The truth of the statement for non-primes is trivial, if it is true for P(i), it is also true for P(i+1) > N > P(i), because the statement for P(i) implies this.