Probability that $x^2+y^2+z^2=0$ mod $p$

Question

This question on MSE asked the following:

"Given $x,y,z \in \mathbb{N},$ find probability that $x^2+y^2+z^2$ is divisible by $7.$"

The OP did not declare the assumed probability model, and was duly criticised for that. On the other hand, it is only natural to assume that $x$, $y$, $z$ are independently uniformly distributed mod $7$.

A case analysis then shows that that the probability in question is ${1\over7}$. This simple result led me to solve the same problem for the primes $p=3$, $5$, $11$, and $13$. In each case I obtained ${1\over p}$ as result. Further experiments showed that the remainder of $s=x^2+y^2+z^2$ mod $p$ is not uniformly distributed mod $p$, but that in any case the probability of $s=0$ mod $p$ is equal to ${1\over p}$ for all $p\leq107$. This leads to the following

Conjecture. Let the integeres $x$, $y$, $z$ be independently uniformly distributed modulo the prime $p$. Then the probability that $s:=x^2+y^2+z^2$ is divisible by $p$ is equal to ${1\over p}$.

Maybe this well known. Otherwise I'd like to see a proof.

Answer 1

Found it. Given odd dimension $n$ and quadratic form $$ f = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2, $$ everything in a finite field with odd number of elements $q,$ the count $$ \#\left(f = b\right) \; = \; q^{n-1} + q^{(n-1)/2} \; \; \chi \left( \; (-1)^{(n-1)/2} \; b a_1 a_2 \ldots a_n\right). $$ At the bottom of page 91 Small points out that $$ \#\left(f = 0\right) \; = \; q^{n-1} . $$ When $b \neq 0$ we need to know what $\chi$ means.

Aah. Page 86, very simple. We have finite field $F$ and element $a.$ First $\chi(0) = 0.$ If $a$ is a nonzero square, $\chi(a)=1.$ If $a$ is nonzero and not a square, $\chi(a)=-1.$

Charles Small, Arithmetic of Finite Fields, Theorem 4.6 on page 91,

Answer 2

For the equation $x^2 + y^2 + z^2 \equiv 0 \pmod{p}$, where $p$ is a prime number, we want to find the probability that this congruence holds for randomly chosen integers $x$, $y$, and $z$ modulo $p$.

Let's consider two cases:

1. If $p = 2$, then any integer squared modulo 2 is either 0 or 1. Therefore, for $x^2 + y^2 + z^2 \equiv 0 \pmod{2}$ to hold, all three variables ($x$, $y$, and $z$) must be congruent to 0 modulo 2. In other words, they must be even. The probability of a randomly chosen integer being even is $\frac{1}{2}$.

2. If $p$ is an odd prime, we can use the Three Squares Theorem. This theorem states that for an odd prime $p$, the equation $x^2 + y^2 + z^2 \equiv 0 \pmod{p}$ has solutions if and only if $p \equiv 1 \pmod{4}$ or $p = 2$. If $p \equiv 3 \pmod{4}$, the equation has no solutions.

So, for odd primes $p$ where $p \equiv 1 \pmod{4}$ or $p = 2$, there are solutions to the equation, and for primes $p$ where $p \equiv 3 \pmod{4}$, there are no solutions.

To calculate the probability, we need to consider the distribution of primes. The density of primes congruent to $1 \pmod{4}$ versus congruent to $3 \pmod{4}$ is roughly equal, and the density of primes is inversely proportional to their size. Therefore, you can approximate the probability as:

• For $p = 2$: Probability is $\frac{1}{2}$.
• For odd primes $p$ where $p \equiv 1 \pmod{4}$: Probability is close to $\frac{1}{2}$ because roughly half of the primes fall into this category.
• For odd primes $p$ where $p \equiv 3 \pmod{4}$: Probability is close to 0 because roughly half of the primes fall into this category.

This is a probabilistic analysis based on the distribution of primes, and it provides a general idea of what to expect when considering random primes.