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We know that if $p$ is a prime congruent to $3 \mod 4$, we cannot represent it as sum of two squares. Is there a positive property of such $p$? That is, do we have any statements that say "$p$ is a prime congruent to $3 \mod 4$ iff $\underline{\mbox{a positive statement}}$ is TRUE in an unique way". For instance, we have "$p$ is a prime congruent to $1 \mod 4$ iff $p=a^2+b^2$ and $|ab|>1$ in an unique way.

Turbo
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    $p$ is congruent to $3 \bmod 4$ iff it remains prime in the Gaussian integers $\mathbb{Z}[i]$. Yeah? – Qiaochu Yuan Dec 11 '14 at 00:52
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    Also, please don't type in all caps like that. It's usually considered rude in an online setting (analogous to yelling). – Qiaochu Yuan Dec 11 '14 at 00:55
  • yeah but you are transferring primality to some where else. Also the statement cannot be given proper meaning to rhyme with uniqueness. – Turbo Dec 11 '14 at 01:04
  • corrected the all caps. – Turbo Dec 11 '14 at 01:05
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    There is nothing. For, say $p \equiv 7 \pmod 8,$ we can say there is a minimal set of automorphism orbits solving $x^2 - 2 y^2 = p.$ – Will Jagy Dec 11 '14 at 01:43

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For example, we can say, a prime is $1,3 \pmod 8$ if and only if there is just one expression $p = x^2 + 2 y^2.$

You get a little flexibility by throwing in indefinite forms: a prime is $1,7 \pmod 8$ if and only if there are just two infinite sequences of expressions $p = x^2 - 2 y^2,$ under the action (and its inverse) $$ \left( \begin{array}{r} x \\ y \end{array} \right) \mapsto \left( \begin{array}{rr} 3 & 4 \\ 2 & 3 \end{array} \right) \left( \begin{array}{r} x \\ y \end{array} \right) $$

The two orbits for the prime $7$ have base points $$ \left( \begin{array}{r} 3 \\ -1 \end{array} \right) $$ and $$ \left( \begin{array}{r} 3 \\ 1 \end{array} \right) $$

Will Jagy
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  • Professor. Thank you for the answer. Could you please elucidate the meaning of "four squareclasses involving primes in the 2-adic numbers"? The matrix $(3 4; 2 3)$.. Is there a keyword for these unimodular matrices? – Turbo Dec 11 '14 at 06:35
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I think I will leave the part about squareclasses alone.

Here is a diagram, of a design due to Conway, that organizes all column vectors $$ \left( \begin{array}{r} x \\ y \end{array} \right) $$ of integers $x,y$ such that $x^2 - 2 y^2 = 1,-1,2,-2,7.$

enter image description here

Conway's book can be downloaded from PDF. There is also a helpful discussion in Stillwell, Elements of Number Theory, particularly pages 87-99, that does a good job of showing how the value of the quadratic form and the $(x,y)$ points with $\gcd(x,y)=1$ match up, as I have illustrated for $x^2 - 2 y^2.$

Will Jagy
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  • Thank you but atleast at the stratospheric level how is this related to $2$-adic numbers? just curious? – Turbo Dec 12 '14 at 10:29