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A generic integer $n = p_1^{a_1} \cdots p_n^{a_n}$ is a square iff the $a_i$ are even, and it is a sum of squares iff the $a_i$ are even for $p_i \equiv 3 \ \mathrm{mod} \ 4$.

How can I deduce the probability of a given integer to be a square? A sum of two squares? (and a sum of three squares?) Even for the first question I am dubious of how I should modelize the problem: do we have a probability $1/2$ for each prime in $n$? but the powers grow very fast, so probably I should normalize in some way...

TheStudent
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    There is nothing like a "random" ineteger in the sense that every integer has the same chance to be chosen. Do you mean the asymptotic probability if we take the range $[1,N]$ and $N$ tends to $\infty$ ? – Peter Mar 25 '22 at 09:19
  • The case of two and three squares : Must there be a representation of exactly this number of squares or are fewer allowed ? – Peter Mar 25 '22 at 09:21
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    The case of one square gives of course an asymptotical probability of $0$. The case of two squares is already more interesting. For three squares , there is a nice classification of the numbers allowing such a representation. – Peter Mar 25 '22 at 09:24
  • See for instance https://mathoverflow.net/questions/205862/sums-of-two-squares-positive-lower-density – Michal Adamaszek Mar 25 '22 at 09:49
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    About $\frac56$ of the first $N$ natural numbers are the sum of three squares. For the sum of two squares, see https://math.stackexchange.com/questions/2799979/how-many-integers-leq-n-are-a-sum-of-two-squares-as-n-to-infty – Henry Mar 25 '22 at 10:10
  • @Peter Yes I suppose the best meaning I can want for such a question is the asymptotic probability, or density, of the set of sums of two squares. – TheStudent Mar 25 '22 at 14:02

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