# When a number $N$ is expressible as a sum of two squares in EXACTLY three ways

Does there exist an $$N$$ such that $$N=x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2\neq x_4^2+y_4^2\qquad \land \qquad \begin{cases}\gcd(x_1,y_1)&=1 \\\gcd(x_2,y_2)&=1\\\gcd(x_3,y_3)&=1 \end{cases}$$ ???

If such a number $$N$$ is expressible as a sum of two squares in exactly $$3$$ and only $$3$$ ways, is it guaranteed that the second condition above FAILs to be satisfied?

My way of deriving such numbers $$N$$:

$$\text{if}\quad N=\prod_{i=1}^k\underbrace{p_i^{\alpha_i}}_{{\text{all} \quad 1 \pmod 4\quad \text{primes}}}\qquad \text{then} \qquad r(N)=\frac 1 2 \prod_{i=1}^k(\alpha_i+1)$$ with $$r(x)$$ being the function which is valued by the number of ways a natural number $$x$$ is expressible as a sum of two squares in distinct ways (without respect to order, or negatives, also not including $$0$$). For only three ways to express $$N$$, I chose $$k=2$$ and let the $$\alpha_i$$ be $$1$$ and $$2$$ only.

Examples, \begin{aligned}325&=5^2\cdot 13 &&\to r(325)=3\\ &=1^2+18^2=6^2+17^2=\boxed{10^2+15^2}\\425&=5^2\cdot 17 &&\to r(425)=3 \\&=\boxed{5^2+20^2}=8^2+19^2=13^2+16^2 \\845&=5 \cdot 13^2&&\to r(825)=3 \\&=2^2+29^2=\boxed{13^2+26^2}=19^2+22^2\end{aligned}

The below is my first try at a parameterization for the case where $$r(N)=3$$.
\begin{aligned}N&=\left[\frac{(r^2-q^2)u-(2qr)v}{r^2+q^2}\right]^2+\left[\frac{(2qr)u+(r^2-q^2)v}{r^2+q^2}\right]^2\\&=\left[u\right]^2+\left[v\right]^2\\&=\left[\frac{(b^2-c^2)u+(2bc)v}{b^2+c^2}\right]^2+\left[\frac{(2bc)u-(b^2-c^2)v}{b^2+c^2}\right]^2\end{aligned}

Derivation, in case of interest:

$$\begin{cases}x_1&=pq+rs \\ y_1&=pr-qs \\ x_2&= pq-rs &&=ab+cd&&&=u \\ y_2&=pr+qs&&= ac-bd&&&=v\\ x_3 & &&=ab-cd \\ y_3& &&=ac+bd \end{cases} \iff \begin{cases}ab+cd-pq+rs&=0 \\ ac-bd-pr-qs&=0\end{cases}$$ $$\iff \left[\begin{array}{llll}b&c&-q&r \\ c&-b&-r&-q\end{array}\right]\cdot \left[\begin{array}{l}a \\ d \\ p \\ s\end{array}\right]=\overrightarrow{0}\implies\left[\begin{array}{llll}b&c&-q&r \\ c&-b&-r&-q\end{array}\right]\sim \left[\begin{array}{llll}1 & 0 & -\frac{bq+cr}{b^2+c^2} & \frac{br-cq}{b^2+c^2} \\ 0&1&\frac{br-cq}{b^2+c^2}&\frac{bq+cr}{b^2+c^2}\end{array}\right]\implies$$ $$a=\left(\frac{bq+cr}{b^2+c^2}\right)p+\left(\frac{cq-br}{b^2+c^2}\right)s=\frac{b(pq-rs)+c(pr+qs)}{b^2+c^2}=\frac{bu+cv}{b^2+c^2}$$ $$d=\left(\frac{cq-br}{b^2+c^2}\right)p-\left(\frac{bq+cr}{b^2+c^2}\right)s=\frac{-b(pr+qs)+c(pq-rs)}{b^2+c^2}=\frac{cu-bv}{b^2+c^2} \implies$$ $$x_3=ab-cd=\frac{1}{b^2+c^2}\left[b(bu+cv)-c(cu-bv)\right]=\frac{(b^2-c^2)u+(2bc)v}{b^2+c^2}\sim y_3=\frac{(2bc)u-(b^2-c^2)v}{b^2+c^2}$$ $$\begin{cases}u&=pq-rs \\ v&=pr+qs\end{cases}\implies \begin{cases}p&=-\frac{qu+rv}{r^2+q^2}\\s&=\frac{ru-qv}{r^2+q^2}\end{cases}$$ $$\implies x_1=\frac{(r^2-q^2)u-(2qr)v}{r^2+q^2},\qquad y_1=\frac{(2qr)u+(r^2-q^2)v}{r^2+q^2}$$

• as long as the number $N$ is odd and not itself a square, your $r(N)$ is correct : in these cases the total number of representations is a multiple of $8$ . Theorem 65 in Dickson, Introduction to the Theory of Numbers (1929) Dec 16, 2020 at 4:07
• When you write, "proportionality between the two numbers," I think what you mean is that the two numbers are not relatively prime, that is, they have a common divisor other than $1$. Is that right? Dec 31, 2020 at 4:11
• And what are your "examples" examples of? They are not examples of $r(n)=3$; are they just examples of computing $r(n)$? Dec 31, 2020 at 4:13
• @GerryMyerson yes Dec 31, 2020 at 5:03
• @GerryMyerson the examples where $r(N) \neq 3$ have more to do with a previous edit of this question.... Dec 31, 2020 at 5:11

If $$N=p^2q$$ where $$p\ne q$$ are both primes congruent $$1\bmod4$$, then there exist unique $$a,b$$ with $$p=a^2+b^2$$, $$a>b>0$$, and $$c,d$$ with $$q=c^2+d^2$$, $$c>d>0$$. Then $$p^2=(a^2-b^2)^2+(2ab)^2$$, and \begin{aligned}N&=\left[(a^2-b^2)c+(2ab)d\right]^2+\left[(a^2-b^2)d-(2ab)c\right]^2\\&=\left[(a^2-b^2)d+(2ab)c\right]^2+\left[(a^2-b^2)c-(2ab)d\right]^2\\&=(pc)^2+(pd)^2\end{aligned}

Note that $$2N$$ has the same number of representations as $$N$$.

Also, if $$p$$ is a prime congruent $$1\bmod4$$, then $$p^4$$ has three representations, e.g., $$5^4=625=25^2+0^2=24^2+7^2=20^2+15^2$$.

EDIT: Let's look at this over the Gaussian integers, $${\bf G}=\{\,a+bi\mid a,b{\rm\ in\ }{\bf Z}\,\}$$. Here are some facts we will use without proof. Proofs can be found all over the web, or in many intro Number Theory and Algebraic Number Theory texts.

1. $$\bf G$$ is an integral domain.

2. The units in $$\bf G$$ are $$\{\,\pm1,\pm i\,\}$$. Two elements are called associates if their quotient is a unit. The conjugate of $$a+bi$$ is $$a-bi$$. The conjugate of $$z$$ is denoted $$\overline z$$.

3. The primes in $$\bf G$$ are $$1+i$$, $$a\pm bi$$ where $$a^2+b^2$$ is a $$1\bmod4$$ prime, each $$3\bmod4$$ prime, and all their associates.

4. $$\bf G$$ is a unique factorization domain. In particular, every $$1\bmod4$$ prime $$p$$ has a unique factorization (up to associates) $$p=(a+bi)(a-bi)$$ where $$a^2+b^2=p$$.

Now, every expression of $$N=r^2+s^2$$ as a sum of two squares corresponds to a factorization of $$N=(r+si)(r-si)$$ as a product of a Gaussian integer and its conjugate. In particular, if $$N=p^2q$$ where $$p=a^2+b^2$$ and $$q=c^2+d^2$$ are distinct $$1\bmod4$$ primes, then $$N=(a+bi)^2(a-bi)^2(c+di)(c-di)$$ and there are three ways to write this as a product of a Gaussian integer and its conjugate:

1. $$N=[(a+bi)^2(c+di)]\ [(a-bi)^2(c-di)]$$

2. $$N=[(a+bi)^2(c-di)]\ [(a-bi)^2(c+di)]$$

3. $$N=[(a+bi)(a-bi)(c+di)]\ [(a+bi)(a-bi)(c-di)]$$

Multiplying everything out, this gives $$N=e\overline e$$ where $$e$$ takes on the values

1. $$(a^2-b^2)c-2abd+((a^2-b^2)d+2abc)i$$,

2. $$(a^2-b^2)c+2abd+(-(a^2-b^2)d+2abc)i$$,

3. $$pc+pdi$$.

And that gives our three ways to write $$N$$ as a sum of two squares.

It's clear that the representation coming from the third option, $$N=(pc)^2+(pd)^2$$, involves numbers that are not relatively prime.

• $\gcd(a,b)=\gcd(c,d)=1$. From that, you can show that only the $(pc)^2+(pd)^2$ is not relatively prime. Everything becomes simpler when analyzed in terms of complex numbers (Gaussian integers). We're looking at the different choices of sign in $(a\pm bi)(a\pm bi)(c\pm di)$. If you don't like zero, then $5^5=55^2+10^2=50^2+25^2=41^2+38^2$. Dec 31, 2020 at 9:07
• The parametrization may be similar to yours, but you have all these quantities $r,q,u,v,b,c$ with no indication of where they come from, whereas I tie $a,b,c,d$ explicitly to the representations of $p,q$ as sums of two squares. Dec 31, 2020 at 9:14
• I've added my derivation to the post. I do like your approach, and your parameterization is likely better, but I just don't see how this parameterization proves that there IS NOT some N with the above second property that every (x,y) pair be mutually prime. Is this the ONLY parameterization possible??? Dec 31, 2020 at 16:05
• If $n=p^2q$ then there will be a representation of $n$ obtained by taking the representation of $q=c^2+d^2$ and multiplying both $c$ and $d$ by $p$. So, you can't avoid having a pair that isn't mutually prime. Similar remarks apply to $n=p^5$ and $n=p^6$. $5^6=117^2+44^2=120^2+35^2=100^2+75^2$. Dec 31, 2020 at 22:38