Answer:
Solution
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Let us assume that there are a finite number of positive primes p
1
, p
2
, , . . . ,p
n
such that p
<p
3
<...<p
Let x=1+p
p
. . . p
We know that p
is divisible by each of p
,p
But x=1+p
is not divisible by any one of p
,...,p
∴ x is a prime number or we can say that it has prime divisors other then p
There exists a positive prime divisor other than p
This contradicts our assumption that there are a finite number of positive primes.
Hence, the number of positive primes is infinite
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Answers & Comments
Answer:
Solution
verified
Verified by Toppr
Let us assume that there are a finite number of positive primes p
1
, p
2
, , . . . ,p
n
such that p
1
<p
2
<p
3
<...<p
n
Let x=1+p
1
p
2
p
3
. . . p
n
We know that p
1
p
2
p
3
. . . p
n
is divisible by each of p
1
,p
2
,p
3
<...<p
n
But x=1+p
1
p
2
p
3
. . . p
n
is not divisible by any one of p
1
,p
2
,...,p
n
∴ x is a prime number or we can say that it has prime divisors other then p
1
,p
2
,...,p
n
There exists a positive prime divisor other than p
1
,p
2
,...,p
n
This contradicts our assumption that there are a finite number of positive primes.
Hence, the number of positive primes is infinite