I There are only finitely many primes

[martinbn]

I just saw this one. If there are finitely many primes, then
##0<\prod_{p}\sin(\frac\pi p)=\prod_p\sin\left(\frac{\pi(1+2\prod_q q)}p\right)=0##

Of course it is in a way just a variation of Euclid's idea, but it is a one liner.

 

[.Scott]

Responding to the title (Thera are only finitely many primes):

If there are a finite number of primes, then take the product of all of them and add 1.
The number you have will be different than all primes and will not contain any of them as a factor.
- reductio ad absurdum

 

[martinbn]

Responding to the title (Thera are only finitely many primes):

If there are a finite number of primes, then take the product of all of them and add 1.
The number you have will be different than all primes and will not contain any of them as a factor.
- reductio ad absurdum

Yes, that is Euclid's proof.

 

[fresh_42]

I like the one-liner with the sine function, but the beauty of Euclid's proof is that you can explain it to kids. It contains elementary proof techniques and requires only basic arithmetic, condensed mathematics.

 

[DaveC426913]

What is the elementary proof that the product of two (or n) primes plus one is, itself, a prime?
(I must have this wrong. 7x11+1=78)
Researching...

 

[fresh_42]

What is the elementary proof that the product of two (or n) primes plus one is, itself, a prime?
(I must have this wrong. 7+11+1=78)

You only have ##7\cdot 11+1=78=6\cdot 13 \,|\,78 =7\cdot 11+1.##

##78## isn't the new prime, but it contains one, ##13,## that has not been on the previous list ##\{7,11\}.## Otherwise, we had a remainder ##1## and ##0## by the division of ##78## by ##13## which cannot be true.

 

[martinbn]

What is the elementary proof that the product of two (or n) primes plus one is, itself, a prime?
(I must have this wrong. 7x11+1=78)
Researching...

It is not always a prime, but any prime divisor of it will be different from the ones you used.

 

[PeroK]

What is the elementary proof that the product of two (or n) primes plus one is, itself, a prime?
(I must have this wrong. 7x11+1=78)
Researching...

There are two slightly different versions of the Euclid proof.

If we assume that we have all the finitely many prime numbers, then the product of all of them plus must also be prime. As this number is not divisible by any prime. Which is a contradiction.

Alternatively, if we have any finite set of prime numbers, then the product of them plus 1 is either prime or divisible by a different prime. In either case, we have an additional prime. Therefore, no finite set of primes is complete. Therefore, the set of primes must be infinite.