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

Dirichlet's theorem

For any two coprime positive integers $$a$$ and $$d$$, there are infinitely many primes of the form $$a + nd$$, where $$n$$ is a non-negative integer.

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

Proof:

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

Note by Paul Ryan Longhas
8 months ago

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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 $$m=n$$. · 7 months, 3 weeks ago