Question

If you can think of anything special to you from the subject of number theory, will you teach me whatever this special thing that you hold dear to your heart is?

Answer 1

or tell me what it is and i can look it up and learn myself

Answer 2

I am quite fond of continued fractions, not much use for anything though...

Answer 3

i'll be taking number theory in 2 weeks, i'll give you weekly reports :)

Answer 4

There is one, part of number theory, that i like, where you can use it to "guess" the numbers on someones personal check

Answer 5

:)

Answer 6

interesting i haven't head of that one
like account number?

Answer 7

yeah

Answer 8

Euler's totient function is neet

Answer 9

ok continued fractions and i will also look at this number guessing thing

Answer 10

i seen alittle bit of euler's totient function

Answer 11

\[\phi(n)=\phi (p q)=(p-1)(q-1)\]

Answer 12

i will look for it in my notes, but in the mean time, The RSA encryption scheme is pretty awesome

Answer 13

i.e creating ciphers

Answer 14

n is a number made up of two prime factors

Answer 15

i love rsa

Answer 16

i did a research on it
thats where i seen the totient thing

Answer 17

of course Fermat's Little Theorem

Answer 18

yep yep

Answer 19

i took a course in number theory like 4 years ago

Answer 20

i like i think it is called the Chinese remainder theorem
it might have been just the euclidean algorithm
i get everything in that category mixed up

Answer 21

but it doesn't really matter since it is basically the same stuff

Answer 22

i mean think you can use the euclidean algorithm to prove the Chinese remainder thm right?

Answer 23

i can't remember what the Chinese remainder them was lol

Answer 24

something about gcd(a,b)=d
then ax+by=d

Answer 25

for integers x and y

Answer 26

but isn't that the Bloxent identity

Answer 27

or whatever it is called

Answer 28

i remeber the pigeon hole Principle used to explain the division algorithm

Answer 29

Pigeonhole is beast >.> love it.

Answer 30

i gonna look up this pigeon junk
never seen that one

Answer 31

http://www.cut-the-knot.org/do_you_know/pigeon.shtml

Answer 32

i can explain if you want

Answer 33

why pigeons?

Answer 34

Surprised no one mentioned congruences...

Answer 35

why not?

Answer 36

why not little joes?
if there are two joes in one hole and that must mean there an empty hole somewhere

Answer 37

lolol

Answer 38

too many joes for one person to handle >.>

Answer 39

thats why we didn't see two joes go in one hole

Answer 40

thank goodness!

Answer 41

there are some pretty cool problems you can do with it.

Answer 42

like what

Answer 43

Like....(drawing junk one sec)

Answer 44

modular arithmetic is what we could use to determine the bank routing number on someones check :)

Answer 45

so how would you find out what my routing number is?

Answer 46

or tell me what my routing number is?

Answer 47

what kindof information do you need?

Answer 48

Okay lets see if i can do this

Answer 49

lol

Answer 50

this is so exciting

Answer 51

so, there are 9 digits in the bank routing number..

Answer 52

correct

Answer 53

so the only example i have written down, is where u am given 8 digits from a bank routing number, would you be willing to give me 8 digits to work with, dosent have to be you actual bank routing number

Answer 54

one sec let me look up something

Answer 55

let me come up with one 211 872 94_

Answer 56

ok

Answer 57

but how do i know any number can't go there

Answer 58

because i will show you the process to find the exact number that goes there, by the way the number is 6

Answer 59

joemath, I'll be taking number theory soon :P I'll hit you up if I need help x.x

Answer 60

the missing number is 6

Answer 61

ok

Answer 62

here is the problem:
You have a unit square (as shown below). Prove that no matter how you place 5 points inside the square, some 2 will be a distance less than
\[\frac{\sqrt{2}}{2}\] apart

Answer 63

Now, in general, i would use letters to represent these numbers. So, what we do is compute an auxiliary number for this bank number

Answer 64

Can't you prove that the distance across the square is sqrt(2)/2 so anything inside of it obviously has to be closer?

Answer 65

the distance across the whole square is just sqrt(2)

Answer 66

You're right, I just realized that hahahaha

Answer 67

There are only 4 corners so if you did that, then one would have to be in the middle for maximum distance ( I think) If you exclude the boundary then it is just "slightly" closer?

Answer 68

we multiply multiply using 7,2,9:
7*2+3*1+9*1+7*8+3*7+9*2+7*9+3*4+9*(unknown number)

Answer 69

Since the distance to the middle is sqrt(2) then half that is sqrt (2)/2 and you're slightly less than that.

Answer 70

now just go with me on this

Answer 71

we simplfy everything, and get : 196+9*(unknown number)

Answer 72

i think i told my professor that when he gave me the problem, and he didnt take it >.< hetold me something along the lines of, "yes, that makes sense, but thats not a proof"

Answer 73

now, we consider this auxilary number modulo 10, which i am sure you know since you are very smart, we consider its remainder when divided by 10

Answer 74

Do you know the proof?

Answer 75

This bank routing number is designed so that this remainder equals zero.

Answer 76

its like you are saying, "if the points were in the corners, then...." which is just one of infinitely many possibilities. A true proof would cover all cases at once.
He got on my case big time lol >.<
yeah i know the proof now.

Answer 77

i didn't know that
thats cool lagrange

Answer 78

So, we now know that the this number(the auxiliary number), must have a zero remainder when divided by 10 or else there is an error

Answer 79

I see what you're saying. But the one example we gave maximizes the distance between them, any other configuration would have them closer. So we're only proving for the "best" case.

Answer 80

and divisiblity by 10 is easy to check right? casue a number is evenly divisible by 10( so it has a remainder of 0) only when the number ends in zero

Answer 81

Personally, i think that too. my professor doesnt think so though lolol. My prof was like, "What if Gauss came along and said he knew of a way to place the points where the statement isnt true. Your proof wont prove him wrong, while a correct proof would."

Answer 82

Harsh.

Answer 83

yeah, hes rough >.< but i learn a lot from him. its worth his bad attitude lol.

Answer 84

now, we have 196+9*(unknown digit)
this number has to have a zero remainder when divided by 10

Answer 85

so you should be able to come up with any real routing number given any random 8 digits but we use mod 10 right always?

Answer 86

yes mod 10 because we have to have a remainder equal to 0

Answer 87

joe your professor sounds mean

Answer 88

i thought u havent had number theory what class is that?

Answer 89

a problem solving class. it covered a wide range of topics. a WIDE WIDE range lol

Answer 90

So,for our auxilary number to have a remainder equal to zero (that is mod 10) we must find a multiple of 9- so that the mystery number( that we are looking for) when multiplied by 9 has to end in a 4, so that the digit 4 when added to the last digit in the other number (196) will produce a number that ends in 0

Answer 91

Now my question to you is: what digit when multiplied by 9 will end in a 4?

Answer 92

6

Answer 93

exactly, do you see it now

Answer 94

9*6=54

Answer 95

if we let that missing number be 4....

Answer 96

yep i will probably attempt to do one of these later

Answer 97

thanks lagrange