Prove that there is no largest prime number
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Let us assume n∈N be the largest prime number.
Then 2,3,5,7,11,...,n be the list of all prime number.
Let K = 2·3·5·7·11···n,(product of all known prime numbers).
Now consider the numbers (K-1) and (K+1).
Clearly, we can see that none of the known prime numbers(as per our assumption) can be a factor of either (K-1) or (K+1).
Hence (K-1) and (K+1) are new prime numbers which are greater that than n.
Proceeding this way, we can always find two new prime numbers.
Which shows that our initial assumption was wrong, hence there exists no largest prime number.
Then 2,3,5,7,11,...,n be the list of all prime number.
Let K = 2·3·5·7·11···n,(product of all known prime numbers).
Now consider the numbers (K-1) and (K+1).
Clearly, we can see that none of the known prime numbers(as per our assumption) can be a factor of either (K-1) or (K+1).
Hence (K-1) and (K+1) are new prime numbers which are greater that than n.
Proceeding this way, we can always find two new prime numbers.
Which shows that our initial assumption was wrong, hence there exists no largest prime number.
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