We all know pancakes and how delicious they can be if prepared properly, and it is only natural wanting to share it with your closest friend. However, cutting a pancake exactly in half, one for you and one for your friend, is not trivial matter. It is even less trivial when you understand that your friend also has a favorite pancake which he wants to share as well, so now we need to double our cutting process. In this post we will show that with a bit of mathematics, not only you can cut both pancakes exactly in half, but you can do it simultaneously with a single straight line.
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board with 15 numbered pieces as in the image below, where the pieces are ordered according to their numbers, except the 14 and 15 pieces which switch positions. Given this board we are allowed to slide into the missing place any one of the pieces next to it, how many times that we want, and we win if we manage to get all the pieces in the right order.
. While this operation is not well defined, and depends on the presentation of the rational numbers and not just their values, we saw that it still has interesting properties, and in particular 
where 
and 
. Moreover, if we start with 
and add their mediants to get 
, and then repeat the process by adding the mediants of each subsequent pair 
, and continue repeating it again and again, then we get all of the rationals in ![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=333333&s=0&c=20201002)
.
and “returns” the wrong way to add them: 
. This story starts from the most elementary mathematics there is, and goes all the way to the most advanced, and on the way it introduces many interesting concepts in mathematics.
is linear, and to prove it they show that 
for all 
. What Blum-Luby-Rubinfeld (BLR) showed that in a sense it is enough to know that one of this conditions (chosen randomly) is true to conclude that all of them are true with high probability. In this post we will try to see what is so important about linearity, and which structures hides in this problem, to make this miracle happen.
and 
are coprime, then there are infinitely many primes 
with 
. And to top it all, we also conjectured that they are evenly distributed among the integers 
which are coprime to 
and to show that it diverges to infinity and therefore there are infinitely many primes. 
coprime integers, there are infinitely many primes -
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