# Quadratic reciprocity for three primes?

The quadratic reciprocity law states that for $$p_1\ne p_2$$ prime, the product $$\left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)$$ takes values $$1$$ or $$-1$$ depending on whether $$p_1$$ and $$p_2$$ satisfy some set of restrictions mod $$4$$.

Is there a "quadratic reciprocity law for three primes"? I suspect that the answer is negative.

Is it true that for any integers $$M>0$$, $$\varepsilon\in\{-1,1\}$$, and $$r_1,r_2,r_3$$ coprime with $$M$$, there exist primes $$p_1\equiv r_1\pmod M$$, $$p_2\equiv r_2\pmod M$$, and $$p_3\equiv r_3\pmod M$$ such that $$\left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_3}\right)\left(\frac{p_3}{p_1}\right) = \varepsilon ?$$

• I suspect you can take $p_3$ arbitrary, say $1\pmod 4$, and the pick $p_1,p_2$ with $p_1p_2=1\pmod p_3$. I will let someone else work outtge details though – Wojowu Jan 15 '20 at 19:59
• @GerhardPaseman: I actually did some computations, which show that $M=48$ does not work: for any given $r_1,r_2,r_3$ co-prime with $48$, and any $\varepsilon\in\{-1,1\}$ there exist primes $p_1,p_2,p_3$ satisfying the condition. – Seva Jan 15 '20 at 20:16
Let me assume $$4\mid M$$. Pick $$p_2,p_3$$ arbitrary satisfying the congruence modulo $$M$$ (they exist by Dirichlet). Take any $$p_1$$ which is congruent to $$r_1\pmod M$$, congruent to $$p_2^{-1}\pmod{p_3}$$, and such that $$\left(\frac{p_1}{p_2}\right)=\varepsilon\cdot(-1)^{\frac{r_1-1}{2}\frac{r_3-1}{2}}$$ (which exists by Dirichlet, CRT, and existence of (non)residues modulo $$p_2$$.) We have $$\left(\frac{p_2}{p_3}\right)\left(\frac{p_3}{p_1}\right)=\left(\frac{p_2}{p_3}\right)\left(\frac{p_1}{p_3}\right)\cdot(-1)^{\frac{p_1-1}{2}\frac{p_3-1}{2}}=\left(\frac{p_1p_2}{p_3}\right)\cdot(-1)^{\frac{r_1-1}{2}\frac{r_3-1}{2}}=\left(\frac{1}{p_3}\right)\cdot(-1)^{\frac{r_1-1}{2}\frac{r_3-1}{2}}=(-1)^{\frac{r_1-1}{2}\frac{r_3-1}{2}}$$ (second inequality follows since $$p_1\equiv r_1,p_3\equiv r_3\pmod 4$$), so multiplying by $$\left(\frac{p_1}{p_2}\right)$$ leaves $$\varepsilon$$.