Show that the set of all prime numbers is infinite.
Answers
Let P be the set of all prime numbers. We takethe negation of the statement“the set of all prime numbers is infinite”, i.e., we assume the set of all prime numbersto be finite. Hence, we can list all the prime numbers as P1, P2, P3,..., Pk (say). Note
that we have assumed that there is no prime number other than P1, P2, P3,..., Pk .
Now consider N = (P1 P2 P3…Pk) + 1 ... (1)
N is not in the list as N is larger than any of thenumbers in the list.N is either prime or composite.If N is a prime, then by (1), there exists a prime number which is not listed.On the other hand, if N is composite, it should have a prime divisor. But none of thenumbers in the list can divide N, because they all leave the remainder 1. Hence, theprime divisor should be other than the one in the list.Thus, in both the cases whether N is a prime or a composite, we ended up with
contradiction to the fact that we have listed all the prime numbers.Hence, our assumption that set of all prime numbers is finite is false.
Thus, the set of all prime numbers is infinite.
the prime numbers are infinite because numbers are infinite
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