Recently, on a mathematics website, you can see from time to time the discussion on how big Ge Liheng's number is.
Ge Liheng's number is a very huge number. It is difficult for ordinary people who have never been exposed to this field to think of integers, which are difficult to be larger than it.
What are the tens of trillions, what are the Gugor, what are the known universes? Every atom is written as a piece of paper. Even every atom is imagined as a small universe and then uses every atom in it to imagine as a small universe...it is difficult to touch its size.
Of course it is not infinite, so if you repeat the steps in..., you can always catch up with it if you repeat them enough times, but the number of repetitions in ellipsis is still a very huge number. To write clearly the number represented by the ellipsis, you still need a similar process to... go through... This ellipsis is smaller than before, but it is still very large...
I was too lazy to copy the definition of it, and I could find it almost by just searching it.
But the problem is, it's big, then what?
If it is just a big one, no matter how big it is, Ge Liheng's number +1 will be larger.
It is called G(64), so is G(65) bigger than it?
If you check its source, you will find that it is an upper boundary estimate.
What is upper boundary estimation?
For example, a certain place needs food, how much does it cost?
accurate number cannot be answered, but ten million tons should be enough, this is the upper boundary estimate.
10 million tons should be enough, and that two trillion tons should be enough. This is still the upper boundary estimate.
replaces every atom of the entire universe with meters, which is of course enough. This is still an estimate from the upper boundary.
So, as an estimate of the upper bound, is the bigger the more meaningful?
Ge Liheng's number needs to solve this problem:
considers an n-dimensional cube, which has a total of 2^n vertices, and connects these vertices to the line segment, and there are a total of 2^(n-1)(2^n-1) line segments.
paints each line segment with blue or red, so is there a surface where all the six line segments are in the same color? In the case of
n=3, a slight change of the dyeing scheme will not be true
Ge Liheng number is that it tells you that as long as n is not less than Ge Liheng number, there must be such a side.
So can n be smaller?
In fact, the current proof can be much smaller than Ge Liheng's number. I guess it is smaller than G(1), but it is still far exceeding astronomical numbers.
However, if we ask from another perspective, isn't a very small n possible?
Ge Liheng has proved that n must be at least 6.
Now it can be confirmed that n must be at least 13.
. That is to say, for this problem, 13 is the lower bound, and a large number far exceeding astronomical numbers is the upper bound.
If you find that the critical situation of this problem is not high in the future, and to the extreme, the answer may be 13, then the upper boundary like Ge Liheng Number will probably appear a bit embarrassing, although it is meaningful as a simple existence.
