abstract algebra - Determine all the representations of the integer $2130797 = 17^2 times 73 times 101$ as a sum of two squares.

Determine all the representations of the integer $2130797 = 17^2 \times 73 \times 101$ as a sum of two squares.

attempt: Suppose $2130797$ is of the form $n = 2^kp_1^{a_1}....p_r^{a_r}q_1^{b_1}...q_s^{b_s}$ . Where $p_1,...,p_r$ are distinct primes congruent to $1$ modulo $4$ and $q_1,....,q_s$ are distinct primes congruent to $3$ modulo $4$. Then $n$ can be written as sum of two squares in $\mathbb{Z}$. Then the number of representations of $n$ as a sum of two squares is $4(a_1 + 1)(a_2+1)...(a_r + 1)$.

Then $2130797 = 17^2 \times 73 \times 101 = (4 + i)^2(4-i)^2(8 + 3i)(8 - 3i)(10 + i)(10 - i)$

So the number of representations of $2130797 = 4(2 + 1)(1 + 1)( 1 + 1) = 48$

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Can someone please help me ? I don't' know how to continue. I am only able to find the number of representations. But I don't know how to start determining the different ways of representing the integer as a sum of squares.

Any feedback would help. Thank you!

Answer

The representations counted by $48$ are not all fundamentally distinct.

First we look at representations where each number is divisible by $17$, say $17a$ and $17b$. To find the two essentially distinct possibilities for $a$ and $b$, (i) multiply $8+3i$ by $10+i$ or (ii) multiply $8+3i$ by $10-i$. We get two numbers $a+bi$. These give the representations $(17|a|)^2+(17|b|)^2$.

Next we look at representations where the numbers are not divisible by $17$. Take $(4+i)^2$ and multiply it by $(8\pm 3i)(10\pm i)$ in all $4$ possible combinations. If a product is $x+iy$, use the representation $|x|^2+|y|^2$.

From these $6$ basic representations, you can get all representations by changing order and/or signs. The full list is not really worth writing down.