# How big really is the new prime?

Hi,
it has been few weeks/days since the last prime was found. It was a huge boom since it happens less and less often as the prime increases and we are getting to even bigger numbers. How big this number actually is? Well in this post I will try to show it, and it is really huge.

I heard some great analogies to 52! (factorial) which is pretty big number. What I decided to do is to do the same thing with the new prime, but oops, this number is way too big.

This is the number which has been found: 274,207,281

It has 22,338,618 million digits, which is way too high to tell you something.

Now imagine that you are counting, every second you add one. How long time would it take you to get to this number? Long time.

More accurately since that is very relative: 6.80348129009742 × 1022338599

This many times of Universes, roughly. What? The number did not even change! Actually there is like 19 less digits.. still not quite good.

So I started to do those analogies for the time written above.

Ok you have a normal can of coca-cola. Such a can has about 330 milliliters. One drop of water (that is actually SI unit) has 0.05 ml. This means that if you would fill it with the rate of one drop per second you would fill it in 6,600 seconds which means almost two hours.

On height, one can has 0.000122 kilometers. Now imagine that if you would fill it you would then put it on one another. You would rise this tower of cans to the height of Sun, that is 150,000,000 kilometers. With filling each of these cans it would take you about 8 trillion seconds [1], this is roughly 253,678 years.

Then you would destroy such tower and take one drop (0.05 ml) from the Atlantic ocean. Atlantic ocean has volume of 323.6 million of cube kilometers. Before you would take another drop of water from the Atlantic you would have to create this tower again.

It would take you about 1.6 sextillion of years to do this thing [2].

Now imagine that after emptying whole Atlantic ocean, you would fill it again (immediately) but you would take one atom from your body (imagine that your body does not change). Then you would fill those cans, stack them to the sun, do that so many times that you would empty Atlantic ocean and do that so many times that you would actually move all atoms from your body to some different place (there are 7*1027 atoms in your body). Too bad, this would take you only 1.2*1049 years.

When I did those analogies I did not really thought about how big the number was going to be so I was a bit dissapointed, at the same time I realised that the prime number is whole next level and I could equal it only with another monstrosity.

Imagine doing all the previous steps and then arranging 3,900,000 numbers into one row. There are 3900000! combinations and you would do all of them, but between each step you would have to do all the previous. Roughly such a thing would take you the same time as counting by one every second to get to the new prime number.

Dragallur

[1]8 trillion is 8,000,000,000,000
[2]1.6 sixtillion is 1,600,000,000,000,000,000,000

If you really want to see the source of the image: here.

# Crazy large numbers

Hi!
Two or three days ago I learned something about large numbers.
I will talk about numbers small, large and extremely huge numbers.

First of all, small numbers are those which we use in normal live. Every day you use them in your math class or if you want to calculate how much money you spent.
For those numbers you dont need any special way to write them they are quite easy.
One Ten Thousand Million Billion Trillion Quadrillion
That is why in which some states write numbers, it is called short scale because in czech we have:
One Ten Thousand Million Milliard Billion Billiard…. (thats translated)
That is called long scale because there are those “illiards”
http://en.wikipedia.org/wiki/Names_of_large_numbers
Here you can find list of numbers and their names.

Large numbers start to create some problems. If you clicked on the link I posted you probably found that after quintillions you are lost and you dont know how the hell you should remember that.
There is system to write numbers like Unvigintillion. That is ten and sixty six zeros. You probably know this because it is used pretty often: 10^66. I wont explain this for people who dont undestand it because it would be even more boring post than it is now.

At one point this is too small and even if you start to create “towers” of exponents it will look like this: 10^10^10^651682138 which is pretty nasty.
(Btw. e+x means that there is some number of numbers after that number, for example: 153,20e+2 = 153,20
25e+16 = 250 000 000 000 000 000)
So what people created are called Knuth´s up-arrows and they look like this: ↑ (alt+24).
So I will do few examples so you know how it works:
2↑2 = 2^2 = 4
4↑3 = 4^3 = 64
5↑2 = 5^2 = 25
Now you dont get it yet but it gets awesome when i add up one arrow: ↑↑
(I will just remind you that when you have more exponents on more exponents you have to go from right)
2↑↑2 = 2^2`^`2 = 16
4↑↑3 = 4↑4^4^4 = 4↑256 = 1.34e+154
5↑↑2 = 5^5`^`5 = 2.9802322e+17
So it means that second number tells us how many times first number will be there
It gets totally crazy with third arrow: ↑↑↑
2↑↑↑2 = 2↑↑2^2 = 2↑↑4 = 2↑2^2^2^2 = 2↑65536 = 2^2^2^2^2^2^2…. 65536 times
4↑↑↑3 = 4↑↑4^4^4 = 4↑↑1.34e+154 = well I hope you get that idea because now it gets like so crazy that I wont continue but if you want to see some other examples go here: http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

Well thats about all hope you get it, if not then ask me below
Dragallur