Does the Equation Have Non-Trivial Solutions in Any P-adic Fields?

[evinda]

Hello! (Wave)

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't, I have to find at which p-adic fields it has no solution.

I used the following theorem:

We suppose that $a,b,c \in \mathbb{Z}, (a,b)=(b,c)=(a,c)=1$.

$abc$ is square-free. Then, the equation $ax^2+by^2+cz^2=0$ has a non-trivial solution in $\mathbb{Q} \Leftrightarrow$

1. $a,b,c$ do not have the same sign.
2. $\forall p \in \mathbb{P} \setminus \{ 2 \}, p \mid a$, $\exists r \in \mathbb{Z}$ such that $b+r^2c \equiv 0 \pmod p$ and similar congruence for the primes $p \in \mathbb{P} \setminus \{ 2 \}$, for which $p \mid b$ or $p \mid c$.
3. If $a,b,c$ are all odd, then there are two of $a,b,c$, so that their sum is divided by $4$.
4. If $a$ even, then $b+c$ or $a+b+c$ is divisible by $8$.
Similar, if $b$ or $c$ even.The first sentence is satisfied.

For the second one:

$$p=3:$$

$$5+x^2(-7) \equiv 0 \pmod 3 \Rightarrow x^2 \equiv 2 \mod 3$$
$$\left ( \frac{2}{3} \right)=-1$$So, we see that the equation hasn't non-trivial solutions in $\mathbb{Q}$.

To check if there is a solution in $\mathbb{Q}_2$, I used the following lemma:

If $2 \nmid abc$ and $a+b \equiv 0 \pmod 4$, then the equation $ax^2+by^2+cz^2=0$ has at least one non-trivial solution in $\mathbb{Q}_2$.

In our case, $a+b=8 \equiv 0 \pmod 4$, so there is no solution in $\mathbb{Q}_2$, right?

For $p=3,5,7$, I used the following lemma:

Let $p \neq 2$ be a prime, $a,b$ and $c$ be pairwise coprime integers with $abc$ square-free and $p \mid a$, and $Q: ax^2+by^2+cz^2=0$ a quadratic form.
Then there is a solution to $\mathbb{Q}$ over $\mathbb{Q}_p$ iff $-\frac{b}{c}$ is a square $\mod p$.

$$\left( -\frac{5}{-7}\right)=\left( \frac{5}{7} \right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_3$.

$$\left( \frac{-3}{-7} \right)=\left( \frac{3}{7} \right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_5$.

$$\left( -\frac{3}{5}\right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_7$.

It remains to check if the equation has non-trivial solutions in $\mathbb{Q}_p, p \neq 2,3,5,7$.

Can we do this, by only using the pigeonhole principle?

Or do we have to apply Hensel's Lemma? If so, how could we do this? (Thinking)

 

[jvicens]

Hi there! It seems like you have done a thorough job in checking for non-trivial solutions in $\mathbb{Q}$ and various p-adic fields. To answer your question, using the pigeonhole principle alone may not be enough to determine if there are non-trivial solutions in all other p-adic fields. Hensel's Lemma, which is a powerful tool in number theory, may be necessary to prove the existence or non-existence of solutions in these fields. It may be worth exploring how Hensel's Lemma can be applied to this particular equation. Additionally, other techniques such as quadratic reciprocity may also be useful in this case. Keep up the good work!