Number Theory-Symmetric Divisibility Conditions
by XmL, May 28, 2014, 5:55 AM
Today on Brilliant a problem like this caught my eyes:
Find all primes
such that
.
As mentioned in the title, this is an example of a problem with "Symmetric Divisibility", and it is also one of my favorite explorative topics in number thoery(although I'm not sure if there's an actually devotion to this topic).
Solution
Note to solution
Another "Symmetric Divisibility" I like is
Find all primes


As mentioned in the title, this is an example of a problem with "Symmetric Divisibility", and it is also one of my favorite explorative topics in number thoery(although I'm not sure if there's an actually devotion to this topic).
Solution
First of all,
must be distinct primes, if not, let's say
, then
a contradiction. Hence WLOG let
.
Now just consider the last two conditions, since
, the two conditions are equivalent to
, note that if we treat
as a constant, then
is definitely boundable. In fact
. Since the second term is less than 0, hence
or
.
Hence our equation is now
. (tip: since we are dealing with primes, a natural instinct is to look at parity to see if
could be used, in this case however it didn't spark anything). Now take a look at our first condition:
, we take mod
of our equation:
, this means
, solving this using factoring gives 




Now just consider the last two conditions, since







Hence our equation is now







Note to solution
the reason that I wrote the tip is because I actually considered it and ended up incorrectly deduce that
, which I noticed while writing this proof. When I was working on this problem, I found that
before the bound on
, knowing this immediately gives you 
As you can see with this example, although the conditions given in the problem are symmetric, you sometimes have to "ruin" it by taking it apart in order to solve it.




As you can see with this example, although the conditions given in the problem are symmetric, you sometimes have to "ruin" it by taking it apart in order to solve it.
Another "Symmetric Divisibility" I like is

This post has been edited 2 times. Last edited by XmL, Aug 12, 2014, 11:18 PM