Question #ca15a

2 Answers
Dec 2, 2017

There are many ways. I'll just show you one.

Explanation:

Imagine that there is a finite number of primes #n#.
The number #n!# will be the product of every number #<=n# , including all prime numbers. Then #n!+1# will not be a product of any number #<=n#, so will be prime. We can continue the reasoning, and always get new prime numbers, so the number o prime numbers is infinite.

Dec 2, 2017

Prove by contradiction:
Considering a finite number of primes

Explanation:

We well solve this via the use of contradiction, we we can let there be a finite number of prime #p# where we can let the be primes;
# p = {p_1,p_2,p_3,...,p_n}#
And we know #p_1p_2p_3...p_n# is certainly not prime as it has factors #p_1 ,p_2,..p_n#

But we can consider #p_1p_2p_3...p_n + 1 #

This on the other hand is prime, as if it were not prime than we could devide it by a smaller prime and yield a natural number, we can prove this by;

let #p_m# be the #m#th prime, for #1##<=##m##<=##n#
then #(p_1p_2p_3....p_n + 1)/p_m# is not natural as this is equal to;
#1/p_m + (p_1p_2...p_n)/p_m#
were we know #(p_1p_2...p_n)/p_m# is natural as #m<=n#
But #1/p_m notin ZZ# for #m >=1#
So hence #1/p_m + (p_1p_2...p_n)/p_m# # notin ZZ#

So hence #1 + p_1p_2...p_n# has no prime factors and hence is prime

Hence for any finite set of primes we can make a new prime, hence there is not finitely many primes

Hence there can't be a finite number of primes, hence proven via contridiction