# When does a prime $p=m^2+n^2$ divide the term $m^3+n^3-4$?

I came across a math olympiad type question that goes like this:

For what primes $p$ will $p=m^2+n^2$ and $p$ divide $m^3+n^3-4$?

I tried a few examples and think that $p=5$ is the only solution but am unable to prove it. The property that $p \equiv 1 \mod 4$ seems to not be helpful so far. Any clues or solutions are welcome.

• You have $m^3+n^3-4$ in the title but exponents are $2$ in the body. Which is correct? (I guess the former) – Wojowu Sep 1 '16 at 18:50
• Note: I reformatted the title but left the body the same (though I am fairly sure you meant to have $m^3+n^3-4$ in the body as well). If that's correct, please edit accordingly. – lulu Sep 1 '16 at 18:51
• $p = 2$ technically works too. – 6005 Sep 1 '16 at 19:11
• Are $m$ and $n$ positive? – 6005 Sep 1 '16 at 19:12
• Just an interesting course of events: $$p=m^2+n^2$$ $$mp=m^3+mn^2$$ $$np=nm^2+n^3$$ $$p(n+m)-(nm^2+mn^2)-4=m^3+n^3-4$$ $$p(n+m)-nm(m+n)-4=m^3+n^3-4$$ $$nm(m+n) \equiv -4 \pmod{p}$$ $$n^2m^2(m^2+n^2+2mn) \equiv 16 \pmod{p}$$ $$2n^3m^3 \equiv 16 \pmod{p}$$ $p=2$-one solution, otherwise: $$n^3m^3 \equiv 8 \pmod{p}$$ – rtybase Sep 1 '16 at 21:45

We can use the Gaussian integers. Assume that $m^2 + n^2$ divids $m^3 + n^3 - 4$.
Note that $m^2 + n^2 = (m + in)(m - in)$, and since $m^2 + n^2$ is prime in $\mathbb{Z}$, both of these are prime in $\mathbb{Z}[i]$. Additionally, by this result, as $m,n$ are relatively prime here, $\mathbb{Z}[i] / (m+in) \cong \mathbb{Z}/(m^2 + n^2) \mathbb{Z}$. So we work in the field $\mathbb{Z}[i] / (m+in)$, getting $$m^3 + n^3 - 4 \equiv (-in)^3 + n^3 - 4 = (i+1)n^3 - 4 \equiv 0 \pmod{m+in}$$ Thus $$(1+i) n^3 \equiv -4 = -(1+i)^4 \pmod{m+in}.$$ If $m = n = 1$, we have a solution $\boxed{p=2}$. Otherwise, $1+i$ is not zero, and we divide by it to get $$n^3 = -(1+i)^3 \pmod{m+in}.$$
So the question now is to solve a cubic in $\mathbb{Z}[i]/(m+in)$. It definitely has one solution, $n \equiv -(1 + i)$. It may have two other soutions, if $1$ has three cube roots in $\mathbb{Z}[i](m+in)$, which happens exactly when $m^2 + n^2 \equiv 1 \pmod{3}$ (see here). If so, then there is a Gaussian integer $\omega$, $\omega \ne 1$, such that $\omega$ and $\omega^2$ are the two nontrivial cube roots of $1$ in $\mathbb{Z}[i] / (m+in)$. Otherwise, let $\omega = 1$. Either way, we have $n \equiv -\omega^a (1+i) \pmod{m+in}$, for $a \in \{0,1,2\}$.
Using the isomorphism between $\mathbb{Z}[i]/(m+in)$ and $\mathbb{Z}[i]/(m-in)$ given by conjugation, of course $n \equiv -(\overline{\omega})^a (1-i) \pmod{m-in}$.
• In your first line, do you mean "divides $m^3+n^3 - 4$"? – Erick Wong Sep 1 '16 at 20:22