Question

@SithsAndGiggles Would you explain me why there are infinite primes

Answer 1

I understood everything here http://www.math.utah.edu/~pa/math/q2.html
But didn't get after But when we divide .....

Answer 2

I'm no number theorist, sorry...

Answer 3

But you are experienced in mathematics than , see that link and i didn;t actually get what is written after
But when we divide...

Answer 4

3*5+1=16 not a prime :P

Answer 5

you can always construct a new number that cannot be factored using existing primes :


number requiring a new prime = \(\large p_1*p_2*\cdots + 1\)

Answer 6

means

Answer 7

for example, assume the only primes are {2,3,5}

number requiring a new prime = 2*3*5 + 1
= 31

Answer 8

3*5+1=16

Answer 9

Clearly "2*3*5+1" is not divisible by any of the existing primes as they leave "1" as remainder - By construction

so you need a new prime to represent this number "2*3*5+1"

Answer 10

yes, can u represent 16 as a product of powers of 3 and 5 alone ?

Answer 11

no i see

Answer 12

Can u explain me the line in this link

http://www.math.utah.edu/~pa/math/q2.html

Answer 13

But when we divide p by pn..........

Answer 14

Suppose there are a total of \(n\) primes :
\[\large p_1, p_2, p_3, p_4, \cdots , p_n\]

then, every number can be expressed as product of these primes, yes ?

Answer 15

@rational sure likes primes^_^

Answer 16

yes

Answer 17

not really haha!

incase you're wondering how on earth every number can be expressed as product of primes, here is a theorem that has a proof :

http://prntscr.com/4gcbkw

Answer 18

It could be i thought , because i just took a few examples in my mind

Answer 19

though it dosen't prove it

Answer 20

so , i assumed it

Answer 21

Since we have assumed that the only primes are the one we listed earlier,
if we could show that there is a number that cannot be expressed using exisitng primes, we will be done with the proof, right ?

Answer 22

so we cookup a number which cannot be expressed using any of the exisitng primes :

\[\large p_1, p_2, p_3, p_4, \cdots , p_n\]

Answer 23

But i want the the statement there to be explained, i got all this

Answer 24

This statement:-
But when we divide p by pn we get a remainder 1 .........................

Answer 25

^ in the link http://www.math.utah.edu/~pa/math/q2.html

Answer 26

and that number is :

\[\large p_1* p_2*p_3*p_4* \cdots *p_n+1\]

consider dividing this number by each of the exisitng primes

Answer 27

whats the remainder when you divide `7*8*9+1` by ` 7` ?

Answer 28

1 i did that orally but check

Answer 29

do division algorithm or whatever they call you get the remainder of 1

Answer 30

you're right !
EVERY existing prime that we had assumed earluer(p1, p2, p3, ... pn) leave a remainder 1

so are we convinced that NONE of the existing primes divide this NEW number evenly ?

Answer 31

next we can see how this observation forces us to include a new prime number to our existing prime numbers

Answer 32

I got the concept right into my brain thank you lol

Answer 33

If we actually see , Euclid's proofs , most of them , had assumptions , that is really somewhat difficult to grasp at first sight

Answer 34

As he(Euclid once said):- There are no proofs without assumptions

Answer 35

yeah thats true :) btw, this particular proof is considered as the most ELEGANT proof in the history of math

Answer 36

go through this paragraph :
http://prntscr.com/4gcggg

Answer 37

Yeah , i know , there are some proofs , in topology , but that is just too much complex
this is relatively simple and straight