adamant's blog

By adamant, history, 19 months ago, In English

Hi everyone!

As I continue working through Project Euler, I want to write a blog about another piece of general mathematical knowledge that is both interesting on its own and might be useful in some problems. Consider the following Diophantine equation:

$$$ x^2 + y^2 = z. $$$

We assume that we're given a specific number $$$z \in \mathbb Z$$$, and we need to check if there are $$$x, y \in \mathbb Z$$$ for which the identity above holds. Then, if such numbers exist we should find them and report. Example of a problem that might need it:

Timus — 1593. Square Country. Version 2. You're given an integer $$$n$$$. Find minimum $$$k$$$ such that $$$n = a_1^2+\dots+a_k^2$$$.

Tl;dr.

Let $$$z = 2^{k_0} p_1^{k_1} \dots p_n^{k_n} p_{n+1}^{k_{n+1}} \dots p_m^{k_m}$$$, where $$$p_1, \dots, p_n$$$ are different prime numbers with remainder $$$3$$$ modulo $$$4$$$, and $$$p_{n+1}, \dots, p_m$$$ are different prime numbers with remainder $$$1$$$ modulo $$$4$$$. Then there are two cases. If any of $$$k_{1}, \dots, k_n$$$ is odd, there are no solutions. Otherwise there is always a solution $$$z = x^2 + y^2$$$ that looks like

$$$ x+ iy = (1+i)^{k_0} p_{1}^{k_{1}/2} \dots p_n^{k_n/2} (x_{n+1}+iy_{n+1})^{k_{n+1}} \dots (x_m + iy_m)^{k_m}, $$$

where $$$i^2=-1$$$ and $$$x_k^2+y_k^2 = p_k^2$$$ for $$$k$$$ from $$$n+1$$$ to $$$m$$$. For each $$$p_k$$$, to find such $$$x_k, y_k$$$ we need to find an integer $$$i$$$ such that $$$i^2 \equiv -1 \pmod{p}$$$, then find a minimum $$$x_k = i y_k \bmod p_k$$$ for $$$1 \leq y_k < \sqrt {p_k}$$$. This is doable in $$$O(\log p_k)$$$.

And if we want to count solutions, their number is given by Jakobi's two-square theorem: The number of ways of representing $$$z$$$ as the sum of two squares is $$$4(d_1(z) - d_3(z))$$$, where $$$d_k(z)$$$ is the number of divisors of $$$z$$$ that have remainder $$$k$$$ modulo $$$4$$$.


From exact equation to $$$\bmod z$$$

First of all, let's solve a bit easier problem. One obvious thing we can do is to take remainder modulo $$$z$$$ on both parts:

$$$ x^2+y^2 \equiv 0 \pmod z. $$$

This relaxed version is equivalent to finding $$$x, y, k \in \mathbb Z$$$ such that

$$$ x^2 + y^2 = kz. $$$

Hmmm... Remainders modulo arbitrary number $$$z$$$ is not the most pleasant thing to work on directly. But remainders modulo prime number $$$p$$$ are usually nice. On the other hand, if $$$p$$$ is some prime factor of $$$z$$$ and there is a solution for $$$z$$$, it means that there will as well be a solution for $$$p$$$ with $$$k=\frac{z}{p}$$$. So, let's assume $$$z$$$ to be prime, for now.

From arbitrary $$$z$$$ to prime $$$p$$$

Now, we have another equation

$$$ x^2 + y^2 \equiv 0 \pmod{p}, $$$

where $$$p$$$ is a prime number. What's good about prime numbers is that remainders modulo prime numbers form a field (i.e. they work very similarly to rationals, and we can expect similar results to hold). For $$$p=2$$$, there is a non-trivial solution $$$x=y=1$$$. What about odd numbers $$$p$$$? There are two cases to consider, as the remainder of $$$p$$$ modulo $$$4$$$ is either $$$1$$$ or $$$-1$$$.

Fermat's theorem on sums of two squares tells us that for an odd prime $$$p$$$, the solution exists if and only if $$$p$$$ has a remainder $$$1$$$ modulo $$$4$$$. Moreover, the sum of two squares theorem tells us that the number $$$z$$$ is expressible as a sum of two squares if and only if its prime decomposition does not have a term $$$p^k$$$, where $$$p \equiv -1 \pmod 4$$$, and $$$k$$$ is odd. Let's find out why.

$$$p \equiv 1 \pmod 4$$$

Of course, it's not yet clear why these two cases are important. Let's assume that there is an integer $$$i$$$ such that

$$$ i^2 \equiv -1 \pmod{p}, $$$

that is there is a remainder modulo $$$p$$$ which behaves very similarly to imaginary unit from complex numbers. Then

$$$ x^2 + y^2 \equiv (x+iy)(x-iy) \pmod p. $$$

This reduces the initial equation to a union of linear equations

$$$ \left[ \begin{array}{ll} x+iy \equiv 0 \pmod p, \\ x-iy \equiv 0 \pmod p. \end{array} \right . $$$

For each $$$y$$$, except $$$y=0$$$, there are $$$2$$$ possible values of $$$x = \pm iy$$$, so there are a total of $$$2p+1$$$ solutions. Noteworthy, it is always possible to find a pair of solutions $$$(x,y)$$$ such that $$$1 \leq x, y < \sqrt p$$$, which means that $$$x^2 + y^2 = p$$$ is satisfied exactly.

How to find it? Find $$$i$$$, and consider the minimum value of $$$ik\bmod p$$$ among $$$1 \leq k < \sqrt p$$$. Due to pigeonhole principle, there will be $$$k_1 \neq k_2$$$ such that $$$i (k_1 - k_2) \bmod p \leq \sqrt p$$$. This is actually very similar to 102354I - From Modular to Rational!

Now, when does such $$$i$$$ exist and how to find it? It is known that remainders modulo $$$p$$$ have a primitive root $$$g$$$ such that its powers from $$$0$$$ to $$$p-2$$$ run through all possible remainders modulo $$$p$$$. Note that for odd $$$p$$$ it always holds that

$$$ -1 \equiv g^{\frac{p-1}{2}} \pmod p. $$$

Then, if such $$$i$$$ exists we should be able to find it from

$$$ i^2 \equiv g^{\frac{p-1}{2}} \pmod p \iff i \equiv g^{\frac{p-1}{4}} \pmod p. $$$

Well, technically $$$-g^{\frac{p-1}{4}}$$$ also can be used as $$$i$$$, but it's not that important. What's important is that it is possible to do as above only when $$$p-1$$$ is divisible by $$$4$$$. In other words, when $$$p \equiv 1 \pmod 4$$$.

$$$p \equiv -1 \pmod 4$$$

Now, let's think about the other case. If there is no such $$$i$$$, we can introduce it! Now, we can formally consider numbers that look like $$$x+iy$$$, where $$$i$$$ is not a remainder modulo $$$p$$$. Numbers of this kind, if treated formally, also form a field. If you're familiar with field theory, I should mention that it is isomorphic to the Galois field $$$GF(p^2)$$$. If you're not familiar with it, ignore what I just wrote.

The thing now is that we can try to find all solutions in this new, extended field. And it reduces to the same union of equations

$$$ \left[ \begin{array}{ll} x+iy \equiv 0 \pmod p, \\ x-iy \equiv 0 \pmod p, \end{array} \right . $$$

so for every $$$y$$$, the only possible solutions are $$$x = \pm iy$$$. The problem is, this time such $$$x$$$ would not be a remainder modulo $$$p$$$, unless $$$y=0$$$. Instead, it will be an "imaginary" solution. So, the only "real" solution is $$$x \equiv y \equiv 0 \pmod p$$$. It means that all solutions to

$$$ x^2 + y^2 = kp $$$

look like $$$x = px'$$$ and $$$y=py'$$$. Thus,

$$$ p^2 x'^2 + p^2 y'^2 = kp. $$$

So, if $$$k$$$ is not divisible by $$$p$$$, there are no solutions. Otherwise $$$k=pk'$$$ reduces it to

$$$ x'^2+y'^2 = k', $$$

after which similar argument could be applied. So, if $$$k'$$$ is divisible by an odd power of such $$$p$$$, there are no solutions. We're only one step away from solving the whole $$$x^2+y^2=z$$$ problem now, assuming that we know the factorization of $$$z$$$.

Back to arbitrary $$$z$$$

Now we need to use one more fact from complex numbers. There, we can introduce a norm

$$$ \|x+iy \| = (x+iy)(x-iy) = x^2+y^2. $$$

Its crucial property for this task is that it is multiplicative, that is

$$$ \|(a+ib)(c+id)\| = \| a + ib \| \cdot \| c + id \| . $$$

This gives the Brahmagupta–Fibonacci identity

$$$ (a^2+b^2)(c^2+d^2)= (ac-bd)^2 + (ad+bc)^2, $$$

from which it follows that if we can represent $$$z$$$ as a product of several several numbers that are expressible as a sum of two squares, we can use the identity above to also express $$$z$$$ as a sum of two squares. In complex number terms, it means that we will find a complex number $$$x+iy$$$ such that $$$\|x+iy\| = z$$$.

Repeating what's written in tl'dr section, let $$$z = 2^{k_0} p_1^{k_1} \dots p_n^{k_n} p_{n+1}^{k_{n+1}} \dots p_m^{k_m}$$$, where $$$p_1, \dots, p_n$$$ are different prime numbers with remainder $$$3$$$ modulo $$$4$$$, and $$$p_{n+1}, \dots, p_m$$$ are different prime numbers with remainder $$$1$$$ modulo $$$4$$$. Then there are two cases. If any of $$$k_{1}, \dots, k_n$$$ is odd, there are no solutions. Otherwise there is always a solution $$$z = x^2 + y^2$$$ that looks like

$$$ x+ iy = (1+i)^{k_0} p_{1}^{k_{1}/2} \dots p_n^{k_n/2} (x_{n+1}+iy_{n+1})^{k_{n+1}} \dots (x_m + iy_m)^{k_m}, $$$

where $$$i^2=-1$$$ and $$$x_k^2+y_k^2 = p_k^2$$$ for $$$k$$$ from $$$n+1$$$ to $$$m$$$. This result is tightly connected to the following ones:

Classification of Gaussian primes. A Gaussian integer $$$a+ib$$$ is prime if and only if either of the following is true:

  • $$$a=0$$$ or $$$b=0$$$ and the non-zero number is prime with remainder $$$3$$$ modulo $$$4$$$,
  • $$$a^2 + b^2$$$ is $$$2$$$ or a prime number with remainder $$$1$$$ modulo $$$4$$$.

The result also allows to factor Gaussian integers by representing their norm as a sum of two squares in the way suggested above.

Jakobi's two-square theorem. The number of ways of representing $$$z$$$ as the sum of two squares is $$$4(d_1(z) - d_3(z))$$$, where $$$d_k(z)$$$ is the number of divisors of $$$z$$$ that have remainder $$$k$$$ modulo $$$4$$$.

If there is a prime divisor with remainder $$$3$$$ mod $$$4$$$, it's $$$0$$$. Otherwise, it is $$$4$$$ times the number of divisors of $$$p_{n+1}^{k_{n+1}} \dots p_m^{k_m}$$$. We may interpret it that the divisor decides how much of multipliers in the product would correspond to $$$x_{k} + iy_k$$$, and how much to $$$y_k + i x_k$$$, after which $$$4$$$ accounts for all the possible ways to multiply $$$x+iy$$$ with $$$1$$$, $$$-1$$$, $$$i$$$ or $$$-i$$$, which accounts for all possible combinations of $$$(\pm x, \pm y)$$$.

And, of course, there are similar results for $$$3$$$ squares and $$$4$$$ squares:

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19 months ago, # |
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what is i % p ?

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    19 months ago, # ^ |
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    Any solution to $$$i^2 \equiv -1 \pmod p$$$. It can be found as $$$i \equiv g^{\frac{p-1}{4}} \pmod p$$$, where $$$g$$$ is a primitive root of $$$p$$$, if $$$p = 4n+1$$$. Otherwise, $$$i$$$ does not exist among remainders modulo $$$p$$$.

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19 months ago, # |
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I proposed a trivial version of this problem on codechef once.

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19 months ago, # |
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I got stuck on problem 66 "Diophantine equation" for a very long time two years ago, because I thought you could solve it without googling for anything, as almost all of the first project euler problems. Definitely couldn't :/. Good topic.

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19 months ago, # |
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For those interested:

Task related to two/three/four-square theorems: BOJ 17633

Harder Task: BOJ 17646

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    19 months ago, # ^ |
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    That's very interesting! I wonder what's the construction for 3 and 4 squares is. Would it be sufficient to take one of squares at random until you get a number that factorizes into smaller number of squares?

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      19 months ago, # ^ |
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      Construction for 1 square is trivial I will go over rest.

      Minimum number of squares needed is 4 : As we know a number cannot be represented as a sum of 3 squares if $$$4^a\left(8b+7\right) = n$$$ for any non-negative integer $$$a,b$$$ so a number that can be expressed as $$$8b+6$$$ has a way to express it as a sum of three powers. Assuming we can find a solution to $$$8b+6=x^2+y^2+z^2$$$ our answer will be $$$\left(x\cdot 2^a\right)^2+\left(y\cdot 2^a\right)^2+\left(z\cdot 2^a\right)^2+\left(2^a\right)^2$$$. Our next step will be finding the solution for 3 squares.

      Minimum number of squares needed is 3 : We can just use randomization like you mentioned above. In my testing generating a random number less than $$$10^{18}$$$ and than generating random squares until we find a number such that we can represent our original number minus the square as sum of two squares it took less than 100 random squares generated for $$$10^{5}$$$ tests I ran.

      I am very curious if there is a solution without randomization.

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19 months ago, # |
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I and Vipin were discussing about Pythagorean triplets yesterday. I remembered a problem where we had to find the number of integer solutions of $$$a + b^2 = c^2$$$ till $$$10^5$$$. Does anyone remember where this problem has appeared?

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19 months ago, # |
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Bro how is your math so good? Did you prepare for some math Olympiad? I am really intimidated by your blogs.

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    19 months ago, # ^ |
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    Please don't be :)

    I don't have that much of Math preparation, really. I just studied some basics in university, and I enjoy reading and/or writing about some stuff I find interesting from time to time.

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18 months ago, # |
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Thanks for another beautiful article!

I have a question regarding implementation part: in tl;dr you wrote

For each $$$p_k$$$, to find such $$$x_k, y_k$$$ we need to find an integer $$$i$$$ such that $$$i^2 \equiv -1 \pmod{p_k}$$$, then find a minimum $$$x_k = i y_k \pmod{p_k}$$$ for $$$1 \le y_k < \sqrt{p_k}$$$. This is doable in $$$\mathcal{O}(\log p_k)$$$

How do you suggest to implement it in the given complexity?

I think I have a way to do it, but it is rather different from yours, so I am very interested, what did you mean.

1) First of all, I would struggle to find a primitive root. According to wikipedia, picking random number and verifying, is it a primitive root or not, takes $$$\mathcal{O}(\log \log p \cdot \log p)$$$ time: $$$\frac{p}{\phi(p - 1)} = \mathcal{O}(\log \log p)$$$.

On the other side, being primitive root is too strong property: for $$$p = 4k + 1$$$ consider any element $$$x \in \mathbb{Z}_p^*$$$. If $$$4 | ord(x)$$$, then $$$x^{\frac{ord(x)}{4}} = \pm i$$$ and $$$\mathbb{Z}_p^*$$$ contains at least $$$\frac{p - 1}{2} = \mathcal{O}(p)$$$ elements of that type (any odd power of a fixed primitive root fits). To avoid direct calculating of $$$ord(x)$$$ one can choose $$$i = x^{\frac{p - 1}{2^k}}$$$ for some $$$k$$$. Therefore, we can find $$$i$$$ with non-deterministic $$$\mathcal{O}(\log p)$$$.

2) However, I am far more interested in this, second, point. Any way of finding primitive root seems to be fast enough in practice.

How do you find such $$$y_k$$$ that minimizes $$$x_k$$$?

I can propose a different way: let $$$a \in \mathbb{Z}$$$ be an element from previous item: $$$a^2 \equiv -1 \pmod{p}$$$. Then one can find $$$gcd(a + i, p)$$$ in gaussian integers $$$\mathbb{Z}[i]$$$. It can be shown that $$$\parallel gcd(a + i, p)\parallel= p$$$:

$$$gcd(a + i, P) \cdot \overline{gcd(a + i, p)} = gcd(a + i, p) \cdot gcd(a - i, p) = gcd(a^2 + 1, p) = p$$$

and you can find $gcd$ in gaussian integers, since it's a euclidian domain. Moreover, each time you replace $$$(a, b) \rightsquigarrow (b, a \% b)$$$ the norm of second term at least halves, so $$$gcd$$$ workd in $$$\mathcal{O}(\log p)$$$.

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    18 months ago, # ^ |
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    For 1), I'd do it in standard way to find primitive root. But I like that your idea avoids computing $$$\operatorname{ord}(x)$$$, which requires knowing the factorization of $$$p-1$$$. On the other hand, raising random numbers to $$$\frac{p-1}{4}$$$ is fine too, as they're likely to be primitive roots.

    For 2), I expected kind of same solution as to Library Judge — Min of Mod of Linear. There are several ways to do it, you can e.g. find minimum element with a Euclidean-like process, or with continued fractions (see the solution to 102354I - From Modular to Rational there).

    But I like your solution more, it is an unexpected result for me that $$$\|\gcd(a+i, p)\| = p$$$.

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      18 months ago, # ^ |
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      Thank you, definitely would read the tutorials you provided.