Dirichlet's theorem
For any two coprime positive integers and , there are infinitely many primes of the form , where is a non-negative integer.
Argument: There are infinitely many prime of the form .
Proof:
Let . For which is prime. Otherwise, if then and . Thus, by Dirichlet's theorem there are infinitely many prime of the form .
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Wait this is still an unsolved problem right? The proof may seem correct because n is used in two different contexts in the expression (n)(n)+1, one n is given/fixed while the other n not. Dirichlet can only tell us that there's infinitely many m such that nm+1 is a prime, but it doesn't guarentee that m=n.
Put n=8 Then from n^2+1 We get 8^2 +1 = 64+1 =65 Which is not prime
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There are infinitely many primes of the form n2+1 doesn't mean that all the numbers of the form n2+1 are primes. For example, there are infinitely many positive integers which can be termed as prime numbers and so there are infinitely many primes. Does this mean that all positive integers are prime?