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I'm reading this paper and I think that the author uses the fact that numbers of form $n^2 + 1$ have only prime factors of form $4m + 1$ (excluding the case when $n$ is odd so $2$ is also a possible prime factor). Why is this so?

As I can see, if $n$ is even then $n^2 + 1 \equiv 1\ (\textrm{mod}\ 4)$ and as long as there is even number of prime factors of form $4m + 3$ that won't be violated.

Marko Bukal
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  • $a(n)=4n^2+1:$ oeis/A053755. The result you refer to was partially proven by Cino Hilliard in 2006 and the proof was corrected by Franklin T. Adams-Watters in 2011. I don't spot how to prove it at a glance and would probably need to find the original paper. – JMoravitz Oct 04 '18 at 15:38
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    @JMoravitz Those are the names of the people discussing the result on OEIS. The result was proven by Gauss and is known as quadratic reciprocity. – Misha Lavrov Oct 04 '18 at 15:42
  • @JMoravitz Please be much more careful with duplicate claims. I reopened it. – Bill Dubuque Oct 04 '18 at 16:57
  • @BillDubuque I still do not see the difference between the two. In this question they ask why odd numbers of the form $n^2+1$ (i.e. occurs when $n$ is even, so for $n=2x$ is of the form $4x^2+1$) only has prime factors of the form $4m+1$ and none in the form $4m+3$. In the post I marked as duplicate it asks why numbers of the form $4x^2+1$ (i.e. odd numbers of the form $n^2+1$) factors only into primes of the form $4y+1$. They seem like identical questions to me, just using $n$'s and $m$'s rather than $x$'s and $y$'s. – JMoravitz Oct 04 '18 at 17:01
  • @BillDubuque if you are convinced the post I marked as duplicate was done so incorrectly, then please tell me why you think that. Otherwise I will surely make the same "mistake" again in the future. – JMoravitz Oct 04 '18 at 17:14
  • @JMoravitz As you can see above, there are prior questions wtth much simpler answers that apply immediately. – Bill Dubuque Oct 04 '18 at 17:21
  • @BillDubuque So you are saying that I was not wrong that the post I linked was a duplicate, you are just saying that there were better candidates. This was not made clear in your original comment. – JMoravitz Oct 04 '18 at 17:26

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