Question

Prove that there are infinite prime numbers

Answer 1

I only have the problem in the Either or Case
The Book stated
"If p is the largest prime
Either (p!+1) is prime Or
(p!+1) is divisible by a number between p and (p!+1) in each case, there is a prime number larger than p)
My question is which is the prime number in the Or case? If it is the dividend then how are we sure that it is a prime number?

Answer 2

And what is the negation of
If x+10<0 then 1<x<2 (That's a made up question)

Answer 3

@phi

Answer 4

To be honest I can't follow that.

Answer 5

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

there's EUclids proof.

Answer 6

To prove there is an infinite numbers of prime numbers
We start with the negation
There exists an integer p, such that p is the largest prime
Now p!+1>p and p!+1 is not divisible by p or any other number less than p

Either p!+1 is prime
Or (p!+1) is divisible by a number between p and (p!+1) in each case, there is a prime number larger than p
In each case, there is a prime number greater than p. Thus the negation is false statement

In the Or case, which is the prime number that is greater than p?

Answer 7

I think the idea is you assume a largest prime "p"
then you multiply *all* the primes from 2,3... p
then add 1
it follows that if you divide that number by any prime, you will get a remainder of 1
i.e. none of the primes divide evenly into this number (which is bigger than p)
that means that either p+1 is prime (it obviously divides evenly into itself)
or there is a prime number bigger than p. either way, p is not the biggest prime.
so we have a contradiciton.

Answer 8

yea - thats the Euclid proof.

Answer 9

Please check the proof I have given

Answer 10

I should write
that means that either p!+1 is prime (it obviously divides evenly into itself)

Answer 11

the point is 2 does not divide evenly into 7
you get 3 remainder 1
or by 3: you get 2 remainder 1

Answer 12

Yeah yeah I got it
BUt please explain the confusion from the proof I've given

Answer 13

can you re-word your question ?

Answer 14

So you understand the proof I've given?

Answer 15

yes, I think so.

Answer 16

Either p!+1 is prime
Or (p!+1) is divisible by a number between p and (p!+1) in each case, there is a prime number (((Which prime number are they talking about here))) larger than p

Answer 17

we don't know which prime they are talking about. It is more a question of logic.

you claim there is a biggest prime p
you create a huge number much bigger than p
you show that *none* of the (supposedly) *only* primes that exist divide evenly into this big massive number.

what do we conclude? either there is a number bigger than p that divides into the massive number (it could happen, right?) (btw, that number would be prime)
or it might be that no number less than the massive number divides evenly into it, in which case the massive number is prime.

either way, there is a prime number bigger than p

Answer 18

Okay let me show you an example
5!+1 =121 which is completely divisible by 11. 11 is larger than 5. But how do we know 11 is a prime (consider 11 as n)

Answer 19

Oh got it

Answer 20

The reason the dividend would be a prime is because it can't be a composite since the p is not divisible by any other number less than p. Thus even the dividend cant have a factor less than p. Right?

Answer 21

if we are careful to only multiply out the primes only i.e. not do 5! which is 1*2*3*4*5
but do 2*3*5

Answer 22

Yes I got it
Okay one more thing
what is the negation of
If x+10<0 then 1<x<2 (That's a made up question)

Answer 23

if A then B is also
not A or B
not (not A or B) is A and not B
the negation of if (x+10<0) then 1<x<2) is
(x+10<0) and not (1 < x <2)

Answer 24

so 1>x>2 right?

Answer 25

not(1 < x <2)
means x is not between 1 and 2
in other words,
x<1 or x>2

Answer 26

you get
(x+10<0) and (1 > x or x>2)

if we simplify:
x < -10 and (x< 1 or x >2)
which simplifies to
x<-10