Bedtime stories
by shiningsunnyday, Apr 21, 2016, 3:37 PM
104 is a great book for bed. I've been reading a lot of the theory and the ideas contained are truly enlightening. Before getting to the math, I'll share a story that happened today
Onto the math!
Proposition 1
Proposition 2
Some good-looking practice problems involving the above I dug up
I question myself, often, why can I not talk to girls? This issue is getting quite serious. Ever since growing up, whenever I'm with a girl, I feel extremely awkward. I mean, you can't talk about guy-stuff with girls, and often times I get much lesser mileage with my jokes with girls. Plus, I know nothing about fashion like those super tiny shorts, nail paint, or shampoo brands, so I can't really connect with them.
Today in fitness class:
Me: God... I'm dead tired... How did my extremely-unfit body... just endure running... a... mile? That's like a new record!
Me: I... must... finish...... this last minute... I... I... can... do... it...
*A hot girl steps up to the treadmill next to mine, gave me a
look*
Me: *Forgets that I've been running this whole time* OMG DID SHE JUST WINK AT ME? WHAT SHOULD I DO? SHOULD I WINK BACK? That would be weird... SHOULD I IGNORE IT? That would be rude... *engages in a deep conversation of self-doubt with myself*
Me: Ok, I've gotta at least look good. *fixes my running posture* *dials up the speed and incline* *gives her an "all-day-every-day" impression of my running skills* *retracts my diaphragm to try to hide some of my fat*
Me *10 mins later and I'm still running, muscles feeling like its about to burst*: Wait, did 10 MINS pass since this hottie came? WHAT JUST HAPPENED? Did she just cast a spell on me?? *Steps off, and basically loses balance and falls the moment I come off cause of exhaustion*
Then I realized, I had been narrating out loud.
This happens pretty much everyday. Whenever I walk past a hot girl, say, I adjust my pacing, *tries to look good*, and my heart begins skipping around. I don't even dare to look at her, as it'll make things awkward.
What's wrong with me?
I keep thinking I just look too bad. Most girls talk a lot to those popular jocks. I guess I'm not one of them. Or is there a trick to be able to approach these mystical creatures? Any advice?
.Today in fitness class:
Me: God... I'm dead tired... How did my extremely-unfit body... just endure running... a... mile? That's like a new record!
Me: I... must... finish...... this last minute... I... I... can... do... it...
*A hot girl steps up to the treadmill next to mine, gave me a
look*Me: *Forgets that I've been running this whole time* OMG DID SHE JUST WINK AT ME? WHAT SHOULD I DO? SHOULD I WINK BACK? That would be weird... SHOULD I IGNORE IT? That would be rude... *engages in a deep conversation of self-doubt with myself*
Me: Ok, I've gotta at least look good. *fixes my running posture* *dials up the speed and incline* *gives her an "all-day-every-day" impression of my running skills* *retracts my diaphragm to try to hide some of my fat*
Me *10 mins later and I'm still running, muscles feeling like its about to burst*: Wait, did 10 MINS pass since this hottie came? WHAT JUST HAPPENED? Did she just cast a spell on me?? *Steps off, and basically loses balance and falls the moment I come off cause of exhaustion*
Then I realized, I had been narrating out loud.
Ok that was a joke. I always talk to myself when no one is around, though. But seriously, this issue is getting serious.
This happens pretty much everyday. Whenever I walk past a hot girl, say, I adjust my pacing, *tries to look good*, and my heart begins skipping around. I don't even dare to look at her, as it'll make things awkward.
What's wrong with me?
I keep thinking I just look too bad. Most girls talk a lot to those popular jocks. I guess I'm not one of them. Or is there a trick to be able to approach these mystical creatures? Any advice?
Onto the math!
Proposition 1
Consider, say we take a complete set of residue classes 
and multiply each one by, say 
, add any integer 
to each one, and we would have another complete set of residue classes! Note that a complete set of residue classes 
is a set of 
, integers, which, when taken , yield each residue of 
exactly once.
The formal way of stating such a finding:
Let
, being a positive integer, and let be an integer.
Consider a complete set of residue classes
Then
for 
is also a complete set of residual classes 
Proof
Corollary
Wilson's Theorem
and multiply each one by, say 
, add any integer 
to each one, and we would have another complete set of residue classes! Note that a complete set of residue classes 
is a set of 
, integers, which, when taken , yield each residue of 
exactly once.The formal way of stating such a finding:
Let
, being a positive integer, and let be an integer.Consider a complete set of residue classes
Then
for 
is also a complete set of residual classes 
Proof
Clearly 
as that would imply 
, contradiction, so indeed the resulting numbers form a complete set of residue classes.
as that would imply 
, contradiction, so indeed the resulting numbers form a complete set of residue classes.Corollary
Setting 
gives us the definition of an inverse, specifically that there exists a number, let's say 
such that 
Also, all such numbers are of a unique residue. If the inverse was not unique, call the second one 
then it would imply 
a contradiction since we assumed they were from different residue classes.
gives us the definition of an inverse, specifically that there exists a number, let's say 
such that 
Also, all such numbers are of a unique residue. If the inverse was not unique, call the second one 
then it would imply 
a contradiction since we assumed they were from different residue classes.Wilson's Theorem
For any prime 
Nice exercise involving our proposition. Note that
all have unique inverses, and they come from the set 
. Because we're excluding 
and 
from our set, we can rule out 
as inverses. This implies that the inverses of are from the same set. Now I wish that we can somehow pair up these elements into pairs of inverses, so that 
So in order to generate that large factorial in our theorem, we would like to pair the numbers in such that their product 
The only problem is what if a number is its own inverse? This would mess up the other pairings. Fortunately, this can't be the case as that would mean 
but the largest 
can be is 
, contradiction. Anyways, this means our pairing works and we get Multiplying both sides by yields the desired.
Nice exercise involving our proposition. Note that
all have unique inverses, and they come from the set 
. Because we're excluding 
and 
from our set, we can rule out 
as inverses. This implies that the inverses of are from the same set. Now I wish that we can somehow pair up these elements into pairs of inverses, so that 
So in order to generate that large factorial in our theorem, we would like to pair the numbers in
such that their product 
The only problem is what if a number is its own inverse? This would mess up the other pairings. Fortunately, this can't be the case as that would mean 
but the largest 
can be is 
, contradiction. Anyways, this means our pairing works and we get Multiplying both sides by yields the desired.Proposition 2
Let be a positive integer and 
Consider the set (
, where, when taken of, will reduce to the set of relatively prime integers to . Call such a set a reduced complete set of residue classes. Prove that 
is also a reduced complete set of residue classes.
Proof
Euler's Theorem!
Fermat's Little
be a positive integer and 
Consider the set (
, where, when taken of, will reduce to the set of relatively prime integers to . Call such a set a reduced complete set of residue classes. Prove that 
is also a reduced complete set of residue classes.Proof
Clearly 
would lead to a contradiction, like Proposition 1. So clearly are of unique residue classes . How can we prove they in fact make up the set of relatively prime integers of 
Assume 
, where 
This means that 
but this is ludicrous as that would imply a factor of 
dividing into 
which are both relatively prime to .
would lead to a contradiction, like Proposition 1. So clearly are of unique residue classes . How can we prove they in fact make up the set of relatively prime integers of 
Assume 
, where 
This means that 
but this is ludicrous as that would imply a factor of 
dividing into 
which are both relatively prime to .Euler's Theorem!
Arguably my favorite theorem of number theory.
By Proposition 2,
By Proposition 2,
Fermat's Little
Proof: Set as a prime number 
By definition of a prime number, 
as a prime number 
By definition of a prime number, 
Some good-looking practice problems involving the above I dug up
Let 
be a prime of the form 
which divides 
for integers 
. Prove that are both divisible by
Let be a prime. Show that there are infinitely many positive integers 
such that 
Find all prime numbers and 
for which 
divides the product 
Let
be a prime, and let m and n be relatively prime integers
such that
Prove that is divisible by .
Let
be an integer. Prove that 
is divisible by if and only if is odd.
be a prime of the form 
which divides 
for integers 
. Prove that are both divisible by Let
be a prime. Show that there are infinitely many positive integers 
such that 
Find all prime numbers
and 
for which 
divides the product 
Let
be a prime, and let m and n be relatively prime integerssuch that
Prove that is divisible by .Let
be an integer. Prove that 
is divisible by if and only if is odd.This post has been edited 5 times. Last edited by shiningsunnyday, Apr 21, 2016, 5:12 PM
Sorry. I find this kinda funny. 
May 2018
April 2018