Question

Show that there are infinitely many positive primes.

Answer 1

Hint: number is a prime number only if it is divisible by $1$ and itself.
Ex: $2,3,5,7...$
For all the numbers listed above there are only two factors that are $1$ and number itself.

Complete step by step solution:
The first prime number is $2$.
If $1$ is added to the first prime number $2+1=3$ , now this is also a prime number.
Similarly, $2,3,5$ are also prime numbers if we are multiplying all these prime numbers and $1$ is added to them then the number obtained is $2\times 3\times 5+1=31$
The number obtained is also a prime number.
Consider few examples to make it clear:
a)$2\times 3\times 5\times 7+1=211$
Since $211$ is divided only by $1$ and itself therefore it is considered a prime number.
b)$2\times 3\times 5\times 7\times 11+1=2311$
The number $2311$is also divisible by one and itself therefore it is also a prime number.
$c)2\times 3\times 5\times 7\times 11\times 13+1=30031$
Since the number $30031$is divisible by only one and itself is a prime number.
$d)2\times 3\times 5\times 7\times 11\times 13\times 17+1=510511$
$510511$ is also a prime number.
Therefore, there are many possibilities of generating a prime number just by adding $1$ to the product of prime numbers.
$2\times 5\times 7\times 11\times 13\times 17\times 19\times .......\infty +1=\text{Prime Number}$
Therefore, there are an infinite number of positive prime numbers.

Note: The prime number is a number that is divisible by $1$ and itself therefore the number should be checked for its factors and then conclusion is to be made.
The composite number is a number that has more than two factors and hence it should be noted that we use the written definition of prime numbers to solve this question.