Answer
The are infinitely many prime numbers.
Step-by-step explanation:
(a)The fact that there are infinitely many prime numbers is well known from the times of Euclid. In his book, The Elements, there is the following proof:
Suppose that there are finitely many prime numbers. Let's order them in a sequence [tex]p_{1}, p_{2},..., p_{n}[/tex]. Define the number
[tex]P=p_{1}\cdot p_{2}\cdot p_{3} \cdots p_{n}+1[/tex].
The fundamental theorem of arithmetic establishes that every natural number greater that 1 is prime or the product of prime numbers. Observe that P is not the product of prime numbers because the remainder when we divide by a prime number is one. Then P must be a prime number, and this is a new primer number, which contradicts the hypothesis that [tex]p_{1},...,p_{n}[/tex] are all the prime numbers. Then there are infinitely many prime numbers.
(b) Observe that the set of prime numbers is an infinite subset of natural numbers. Then we can order the prime numbers in a infinite sequence [tex]p_{1},p_{2}, p_{3},...[/tex]. Then there is a one-to-one correspondence between prime numbers and natural numbers. Then, there are as many prime numbers as natural numbers.
