Is it already known that there are infinitely many primes of the form $2n^2 + 2n + 1$? I was searching online for any articles about it but I can't find any so I suppose this is still unknown. I found that there are infinitely many such primes and one result is that there are infinitely many primes of the form $4t_n + 1$ where $t_n$ are triangular numbers.
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It might be worth noting that $2n^2+2n+1=n^2+(n+1)^2$ is the sum of two consecutive perfect squares. – barak manos Nov 07 '16 at 08:24
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Yes indeed that's how I actually found it first. – unknownMe Nov 07 '16 at 08:25
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3This paper claims that no polynomial of degree 2 or higher is known to represent infinitely many primes: https://math.byu.edu/~lzhao/Presentations/primequadprog1.pdf But not for a lack of 94 years of trying. So your result might be met with skepticism. Would you consider posting a clear argument? – 2'5 9'2 Nov 07 '16 at 08:26
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1Related MO topic: Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers – Bart Michels Nov 07 '16 at 08:28
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ok , yeah i got it – unknownMe Nov 07 '16 at 08:31
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The proof is actually can be easily understood. I will try to post it here. – unknownMe Nov 07 '16 at 08:32
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See also this answer, and the links at "Related". – Dietrich Burde Nov 07 '16 at 08:57
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Definitely out-of-reach. $$p=2x^2+2x+1\quad\Longleftrightarrow\quad 2p-1 = (2x-1)^2 $$ hence you claim is equivalent to
There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is twice a prime
but
There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is a prime
is still a conjecture, namely Landau's conjecture. The closest theorem we have at the moment is due to Iwaniec and Friedlander: there are an infinite number of primes of the form $a^2+b^4$.
Jack D'Aurizio
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1there was indeed an error in my proof. I thought and assumed every primitive pythagorean triple is unique*** – unknownMe Nov 17 '16 at 08:46