Answer:
a) There are infinite prime numbers, b) All prime numbers are also abundant numbers
Step-by-step explanation:
To prove a) let's first prove that if n divides both integers A and B then also divides the difference A-B
If n divides A and B, there are integers j, k such that
A = nj and B= nk,
So
A-B= nj - nk = n(j-k)
But j-k is also an integer, which means that n divides also A-B
Now, to prove that there are infinite prime numbers , we will proceed with Reductio ad absurdum.
We will suppose that there are only a finite number of primes and then arrive to a contradiction.
Suppose there are only n prime numbers,
{p1,p2,... pn}
then take P=p1.p2...pn the product of all of them
and consider P+1
If P+1 is prime the proof is complete for P+1 is not in the list.
if P+1 is not prime then by the Fundamental Theorem of Arithmetic there is a prime in the list that must divide P+1, let's say pk
Then pk also divides P+1-P=1 which is a contradiction because no prime divides 1.
b) To prove this, recall that an abundant number is a number for which the sum of its proper divisors is greater than the number itself.
Given that a prime number P is only divided by P and 1, the sum of its divisors is P+1 which is greater than P. So P is abundant
