Is this proof of 4th Landau's problem correct?

Dirichlet's theorem

For any two coprime positive integers aa and dd, there are infinitely many primes of the form a+nda + nd, where nn is a non-negative integer.

Argument: There are infinitely many prime of the form n2+1n^2 + 1.

Proof:

Let n>0n > 0. For n=1,n2+1=2n = 1, n^2 + 1 = 2 which is prime. Otherwise, if n2+12n^2 + 1 \neq 2 then n2+1=(n)(n)+1n^2 + 1 = (n)(n) + 1 and gcd(n,1)=1\gcd(n,1) = 1. Thus, by Dirichlet's theorem there are infinitely many prime of the form (n)(n)+1(n)(n) + 1.

#NumberTheory

Note by Paul Ryan Longhas
5 years ago

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Comments

Wait this is still an unsolved problem right? The proof may seem correct because nn is used in two different contexts in the expression (n)(n)+1(n)(n)+1, one nn is given/fixed while the other nn not. Dirichlet can only tell us that there's infinitely many mm such that nm+1nm+1 is a prime, but it doesn't guarentee that m=nm=n.

Xuming Liang - 5 years ago

Put n=8 Then from n^2+1 We get 8^2 +1 = 64+1 =65 Which is not prime

Aaryan Saha - 5 years ago

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There are infinitely many primes of the form n2+1n^2+1 doesn't mean that all the numbers of the form n2+1n^2+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?

Tapas Mazumdar - 4 years, 1 month ago
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