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We want to find the numbers that divide natural numbers in the form of $n^2+1$ and solve for their natural density.

Using Wolfram Mathematica, I found divisors from $n=0$ to $1000000$ and eliminated repeated divisors. Here is the list

$$\left\{1,2,5,10,13,17,25,26,29,34,37,41,50,53,58,61,65,73,74,82,85,89,97,101,106,109,113,122,125,130,137,145,146,149,157,169,170,173,178,181,185,193,194,197,202,205,218,221,226,229,233,241,250,257,265,269,274,277,281..... \right\}$$

After looking at the list extensively my guess is the density is zero.

Is there a mathematical way of finding this without computer programming?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Aloizio Macedo Mar 28 at 12:44
  • $\begingroup$ @AloizioMacedo Why is my question still unclear? $\endgroup$ – Arbuja Mar 28 at 21:15
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    $\begingroup$ The divisors of $n^2+1$ are precisely the numbers not divisible by four and with no prime factor one less than a multiple of four. These are tabulated at oeis.org/A008784 and there are links there which may lead to an answer to your question. $\endgroup$ – Gerry Myerson Mar 29 at 1:52
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    $\begingroup$ I would suggest eliminating your initial general question, which is hopelessly broad. The specific question about numbers of the form $n^2+1$ is a perfectly good question though. $\endgroup$ – Eric Wofsey Mar 29 at 3:52
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The set in question, call it $A$, is a proper subset of the set, call it $B$, of numbers which can be written as a sum of two squares (the difference between the two sets is that $B$ is the set of all numbers of the form $2^rPQ^2$ where $r$ is a nonnegative integer, the prime factors of $P$ are all $1\bmod4$, and the prime factors of $Q$ are all $3\bmod4$; $A$ is the same, but with $r\le1$ and $Q=1$).

$B$ has density zero (and so, a fortiori, $A$ has density zero). This is discussed at https://mathoverflow.net/questions/205862/sums-of-two-squares-positive-lower-density and at the links to be found there. It goes back to Landau.

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As I understand it, you claim that every divisor of a number of the form $n^2+1$ belongs to one of some set of residue classes (mod 72).

This appears to be true. For instance if any multiple of 3 were to divide $n^2+1$ then we would have $n^2+1 \equiv 0 \pmod{3}$ and in turn $n^2 \equiv -1 \pmod{3}$. But -1 is not a quadratic residue (mod 3), so we can rule out classes like 72m, 72m+3, etc. as potential divisors of $n^2+1$.

I didn't verify your list, but clearly some such list exists.

If you wanted to go from here to calculating the asymptotic density, you'd need to prove something stronger, say that every (perhaps sufficiently large) number belonging to such a class divides some number of the form $n^2+1$.

UPDATE

In fact let's consider that question. Say you have some number $72m + r$ and claim it divides a number of the form $n^2 + 1$. That is, for some k we have $(72m + r)k = n^2 + 1$. Then $-1 + (72m+r)k$ is a perfect square for some $k>0$. But this is the same as $-1$ being a quadratic residue $\pmod{72m+r}$.

My elementary number theory is rusty, but I think that's equivalent to every odd prime divisor of $72m+r$ being of the form $1 \pmod{4}$. So you could try and see whether this applies to all numbers in such a class.

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    $\begingroup$ It is a valiant effort to make sense of the Question, but the considerations you give do not help in determining "density". Showing that certain residues mod $72$ are possible as divisors of numbers of the form $n^2+1$ does not by itself guarantee all such residues as asymptotically equally likely. I applaud your effort to reason on this, but I'm voting to close the Question as the OP has twice asked pretty much the same ill-defined problem. $\endgroup$ – hardmath Mar 26 at 18:35
  • $\begingroup$ @hardmath lulu in the discussion, understands my question. What is still unclear? $\endgroup$ – Arbuja Mar 28 at 22:44
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    $\begingroup$ @Arbuja: With more recent edits to your Question, I'm now able to upvote both it and Daniel's Answer. $\endgroup$ – hardmath Mar 30 at 0:31

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