What does it mean for an integer $n$ to have $k$ expressions as $x^2 + y^2$?

Question

I need to write a Maple procedure to find the first integer $n$ with $k$ representations as $x^2 + y^2 = n$. I do not really understand what this statement means

Answer 1

You’re to write a procedure that, when given $k$, finds the smallest integer $n$ that can be written as the sum of two square in $k$ different ways. For $k=2$, this number is $50$: $50=5^2+5^2=7^2+1^2$. The next such number is $65=7^2+4^2=8^2+1^2$, but you don’t want it: it’s not the smallest such number.

Answer 2

A related concept to what you are being asked to write a program for is that of Taxicab Numbers.

The $n^{th}$ taxi cab number, labled as $\text{Ta}(n)$ is the smallest number that can be expressed in $n$ different ways as a sum of two positive cubes. I.e. the first integer $k$ to have $n$ representations as $x^3+y^3=k$.

The first four taxicab numbers are as follows:

$\begin{matrix} Ta(1) = & 2 & =1^3+1^3\\ Ta(2) = & 1729 & =1^3 + 12^3\\ & & = 9^3 + 10^3\\ Ta(3) = & 87539319 & =167^3+436^3\\ & & =228^3 + 423^3\\ & & =255^3 + 414^3\\ Ta(4) = & 6963472309248 & =2421^3 + 19083^3\\ & & = 5436^3 + 18948^3\\ & & = 10200^3 + 18072^3\\ & & =13322^3 + 16630^3 \end{matrix}$

In much the same way, you are being asked to write a program which finds the smallest number which can be written in $k$ different ways as a sum of squares. (note, we do not consider $9^3+10^3$ different than $10^3 + 9^3$. As such, you may assume without loss of generality that $x\leq y$ in your program to avoid accidentally counting the "same" situation more than once).