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The Ultimate Guide to Solving Pythagorean Triplets with Arm64 Assembly

Finding a Pythagorean triplet in Arm64 assembly is a classic challenge that tests your understanding of low-level logic, register management, and arithmetic operations. This guide breaks down the problem, provides a detailed walkthrough of a complete assembly solution, and explores optimization strategies to turn you into a low-level programming expert.

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The Challenge from the Triangle Tinkerer

Imagine you're a renowned problem-solver, known for cracking complex computational puzzles. One evening, a cryptic message arrives from an inventor known only as the "Triangle Tinkerer." He's building a revolutionary device that relies on the precise geometric properties of right-angled triangles, and he's hit a roadblock.

His letter explains that the device's core mechanism requires integer-sided triangles where the sum of the sides equals a specific perimeter, `N`. He needs you to find all sets of three integers `{a, b, c}` that satisfy two critical conditions: they must form a Pythagorean triplet (a² + b² = c²) and their sum must equal `N` (a + b + c = N). The catch? The calculations must be performed with maximum efficiency, directly on the processor metal. He needs a solution in Arm64 assembly.

This isn't just an abstract math problem; it's a test of your ability to translate high-level logic into the raw, powerful instructions of a modern CPU. You feel the familiar thrill of a challenge. Let's dive in and build the solution he needs.

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What Exactly is a Pythagorean Triplet?

A Pythagorean triplet is a set of three positive integers, let's call them a, b, and c, that satisfy the famous Pythagorean theorem. The relationship is defined by the equation:

a² + b² = c²

For these numbers to form a valid, non-degenerate triangle, they must also adhere to a specific order, conventionally written as:

a < b < c

The most iconic example, often the first one taught in schools, is the set `{3, 4, 5}`. Let's verify it:

• 3² + 4² = 9 + 16 = 25
• 5² = 25

Since 25 = 25, the condition holds. This problem, drawn from the exclusive kodikra.com learning path, adds another layer of complexity: the sum of these three numbers must equal a given integer `N`. For our `{3, 4, 5}` example, the sum is `3 + 4 + 5 = 12`. So, if `N` were 12, this would be a valid solution.

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Why Solve This in Arm64 Assembly?

In a world dominated by high-level languages like Python and JavaScript, you might wonder why anyone would tackle this problem at such a low level. Writing in assembly is an exercise in precision and a direct conversation with the processor. It's about trading ease-of-use for ultimate control and performance.

The Strategic Advantages of Assembly

Understanding assembly is a superpower for a serious developer. It allows you to see past the abstractions of high-level languages and comprehend what your code is *actually* doing. This knowledge is invaluable for performance tuning, debugging complex issues, reverse engineering, and working on systems where every clock cycle and byte of memory counts, such as in operating systems, device drivers, and embedded systems.

Pros	Cons / Risks
Maximum Performance: Direct control over CPU instructions can lead to highly optimized code that outperforms compiler-generated binaries.	High Complexity: Assembly is verbose, hard to read, and requires meticulous management of memory and registers.
Hardware Intimacy: You learn the CPU architecture, its registers, and its instruction set, providing a deep understanding of how computers work.	Not Portable: Code written for Arm64 will not run on x86 or other architectures without a complete rewrite.
Minimal Footprint: Assembly code produces the smallest possible executables, which is critical for embedded systems with limited memory.	Slow Development: Tasks that are a single line in a high-level language can take dozens of instructions in assembly.
Foundation for Systems Programming: Essential knowledge for anyone working on kernels, compilers, or security research.	Error-Prone: Simple mistakes in register allocation or memory access can lead to segmentation faults and unpredictable behavior.

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How to Strategize a Solution: The Brute-Force Approach

Before writing a single line of assembly, we need a solid algorithm. The problem gives us two equations and one inequality for a given sum `N`:

1. a + b + c = N
2. a² + b² = c²
3. a < b < c

A straightforward, if not the most efficient, way to solve this is through brute force. We can iterate through all possible values for a and b and then check if they satisfy the conditions. Since a < b < c, we know that a must be the smallest side. The smallest possible triplet is `{3, 4, 5}`, so a can start at 1 (or 3, to be more precise) and go up. What's the upper limit for a?

If a < b < c, then a + a + a < a + b + c, which means 3a < N, or a < N/3. This gives us a sensible range for our first loop.

Similarly, for our second loop, b must be greater than a. Its upper limit is constrained by a + b + b < a + b + c, which simplifies to a + 2b < N, or b < (N-a)/2. This gives us the bounds for our nested loops.

The Algorithmic Flow

Our algorithm will look like this:

1. Start an outer loop for a from 1 up to N/3.
2. Start an inner loop for b from a + 1 up to N/2.
3. Inside the inner loop, calculate c using the sum equation: c = N - a - b.
4. Check if the order is correct: b < c. If not, we can often break the inner loop because subsequent b values will only make c smaller.
5. If the order is correct, check the Pythagorean condition: does a² + b² = c²?
6. If both conditions are met, we have found a valid triplet. Store `{a, b, c}` and increment our solution counter.
7. Continue until all loops are exhausted.

This nested loop structure is the core logic we will translate into Arm64 assembly. Here is a visual representation of this flow:

    ● Start (Input: N)
    │
    ▼
  ┌─────────────────┐
  │ Loop 'a' from 1 │
  │   to N/3        │
  └────────┬────────┘
           │
           ▼
         ┌─────────────────┐
         │ Loop 'b' from   │
         │   a+1 to N/2    │
         └────────┬────────┘
                  │
                  ▼
            ┌────────────────┐
            │ c = N - a - b  │
            └────────────────┘
                  │
                  ▼
            ◆ b < c ?
           ╱         ╲
      Yes │           │ No
          ▼           │
    ◆ a²+b² == c² ?   │
     ╱         ╲      │
Yes │           │ No  │
    ▼           │     │
┌───────────┐   │     │
│ Store     │   │     │
│ {a, b, c} │   │     │
└───────────┘   │     │
    │           │     │
    └─────┬─────┴─────┘
          │
          ▼
     [Continue Loops]
          │
          ▼
       ● End

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The Code Walkthrough: From Logic to Arm64 Instructions

Now, let's dissect the provided Arm64 assembly solution from the kodikra.com module. This code implements the brute-force strategy we just outlined. Understanding this code requires familiarity with the Arm64 calling convention, where the first few arguments to a function are passed in registers x0 through x7, and the return value is placed in x0.

The function signature in C would look like this:

size_t triplets_with_sum(uint64_t n, uint64_t* a, uint64_t* b, uint64_t* c);

This means:

• x0: Contains the sum n.
• x1: Pointer to an array to store the a values.
• x2: Pointer to an array to store the b values.
• x3: Pointer to an array to store the c values.

The function should return the number of triplets found in register x0.

Register Allocation Plan

Good assembly programming starts with a clear plan for what each register will hold. Let's map them out:

• x0: Initially holds n, but will be used for the return count of triplets.
• x1: Pointer to the results array for a.
• x2: Pointer to the results array for b.
• x3: Pointer to the results array for c.
• x4: A copy of the input sum n (since x0 will be repurposed).
• x5: The loop variable for a.
• x6: The loop variable for b.
• x7: The calculated value for c.
• x8: The return count of triplets (we'll use this as our counter and copy to x0 at the end).
• x9, x10, x11, x12: Temporary registers for intermediate calculations.

The Full Assembly Code

.text
.globl triplets_with_sum

/* extern size_t triplets_with_sum(uint64_t n, uint64_t* a, uint64_t* b, uint64_t* c); */
triplets_with_sum:
    mov     x8, #0          // x8 = 0 (number of triplets found)
    mov     x4, x0          // x4 = n (our sum)
    mov     x5, #0          // x5 = a = 0 (will be incremented to 1 first)

    // Outer loop for 'a'
.L_loop_a:
    add     x5, x5, #1      // a++

    // Optimization: Check if 3*a >= n. If so, no solution is possible.
    // 3*a = a + a + a. If a < b < c, then 3a < a+b+c = n.
    add     x9, x5, x5      // x9 = 2*a
    add     x9, x9, x5      // x9 = 3*a
    cmp     x9, x4          // Compare 3*a with n
    b.ge    .L_return       // If 3*a >= n, exit

    mov     x6, x5          // x6 = b = a (will be incremented to a+1 first)

    // Inner loop for 'b'
.L_loop_b:
    add     x6, x6, #1      // b++

    // Calculate c = n - a - b
    add     x9, x5, x6      // x9 = a + b
    sub     x7, x4, x9      // x7 = c = n - (a + b)

    // Condition: b < c?
    cmp     x6, x7          // Compare b with c
    b.ge    .L_loop_a       // If b >= c, 'c' will only get smaller, so break inner loop and increment 'a'

    // Pythagorean check: a*a + b*b == c*c
    mul     x9, x5, x5      // x9 = a*a
    mul     x10, x6, x6     // x10 = b*b
    mul     x11, x7, x7     // x11 = c*c
    add     x12, x9, x10    // x12 = a*a + b*b
    cmp     x12, x11        // Compare (a*a + b*b) with c*c
    b.ne    .L_loop_b       // If not equal, continue inner loop

    // Found a triplet! Store the results.
    // We need to calculate the memory offset. Since each value is 8 bytes (uint64_t),
    // the offset is count * 8.
    lsl     x9, x8, #3      // x9 = count * 8 (logical shift left by 3 is multiply by 8)
    str     x5, [x1, x9]    // Store a at address (base_a + offset)
    str     x6, [x2, x9]    // Store b at address (base_b + offset)
    str     x7, [x3, x9]    // Store c at address (base_c + offset)

    add     x8, x8, #1      // Increment triplet count
    b       .L_loop_b       // Continue inner loop for b

.L_return:
    mov     x0, x8          // Move final count to return register x0
    ret                     // Return from function

Detailed Instruction Breakdown

1. Initialization

triplets_with_sum:
    mov     x8, #0          // Initialize triplet count to 0
    mov     x4, x0          // Copy n from x0 to x4 for safekeeping
    mov     x5, #0          // Initialize a = 0

The function begins by setting up its state. The triplet counter (x8) is zeroed. The input sum n is moved from the argument register x0 to a general-purpose register x4. This is crucial because x0 must be used for the return value at the end. The loop variable a (x5) is initialized.

2. The Outer Loop (`.L_loop_a`)

.L_loop_a:
    add     x5, x5, #1      // a++

    add     x9, x5, x5      // x9 = 2*a
    add     x9, x9, x5      // x9 = 3*a
    cmp     x9, x4          // Compare 3*a with n
    b.ge    .L_return       // If 3*a >= n, exit function

    mov     x6, x5          // Initialize b = a for the inner loop

This is the start of the loop for a. First, a is incremented. Then, a vital optimization is performed: we check if 3*a >= n. Since we require a < b < c, the smallest possible sum for a given a is slightly more than 3*a. If 3*a is already greater than or equal to n, no solution is possible with this a or any larger a, so we can exit the function entirely. Finally, b (x6) is initialized to the current value of a before its own loop starts.

3. The Inner Loop (`.L_loop_b`)

.L_loop_b:
    add     x6, x6, #1      // b++

    add     x9, x5, x6      // x9 = a + b
    sub     x7, x4, x9      // x7 = c = n - (a + b)

    cmp     x6, x7          // Compare b with c
    b.ge    .L_loop_a       // If b >= c, break inner loop

Here, we increment b, ensuring it's always at least a + 1 on the first run. We then calculate c using the sum identity: c = n - (a + b). The next check is critical for efficiency. We compare b and c. If b is already greater than or equal to c, then any further increase in b will only make c smaller, violating the b < c rule. Therefore, we can break out of the inner loop and jump back to the outer loop to increment a.

4. The Pythagorean Check

This is the mathematical core of the solution. The flow is straightforward: calculate the squares and compare them.

    ● Start Check
    │
    ▼
  ┌────────────┐
  │ x9  = a * a  │
  └─────┬──────┘
        │
        ▼
  ┌────────────┐
  │ x10 = b * b  │
  └─────┬──────┘
        │
        ▼
  ┌────────────┐
  │ x11 = c * c  │
  └─────┬──────┘
        │
        ▼
  ┌─────────────────┐
  │ x12 = x9 + x10    │
  │ (a² + b²)         │
  └─────────┬─────────┘
            │
            ▼
      ◆ x12 == x11 ?
     ╱              ╲
Yes │                │ No
    ▼                ▼
[Store Triplet]    [Continue Loop]

    mul     x9, x5, x5      // x9 = a*a
    mul     x10, x6, x6     // x10 = b*b
    mul     x11, x7, x7     // x11 = c*c
    add     x12, x9, x10    // x12 = a*a + b*b
    cmp     x12, x11        // Compare (a*a + b*b) with c*c
    b.ne    .L_loop_b       // If not equal, jump to the next iteration of the inner loop

We use the mul instruction to compute the squares of a, b, and c, storing them in temporary registers. Then we sum a² and b². The cmp instruction compares this sum with c². If they are not equal (b.ne), it means this is not a Pythagorean triplet, and we branch back to the start of the inner loop to try the next value of b.

5. Storing a Valid Triplet

    lsl     x9, x8, #3      // x9 = count * 8 (calculate memory offset)
    str     x5, [x1, x9]    // Store a
    str     x6, [x2, x9]    // Store b
    str     x7, [x3, x9]    // Store c

    add     x8, x8, #1      // Increment triplet count
    b       .L_loop_b       // Continue inner loop

If the code reaches this point, a valid triplet has been found. We need to store it in the arrays provided as pointers in x1, x2, and x3. To do this, we calculate the memory offset. Since each number is a uint64_t, it occupies 8 bytes. The offset is simply the number of triplets found so far (x8) multiplied by 8. A fast way to multiply by 8 is a logical shift left by 3 bits (lsl x9, x8, #3). The str (store register) instruction then writes the values of a, b, and c to the correct memory locations. Finally, we increment our success counter x8 and branch back to continue the inner loop, as there might be more solutions.

6. Return

.L_return:
    mov     x0, x8          // Move final count to return register x0
    ret                     // Return from function

This is the exit point. The final count of triplets stored in x8 is moved to the designated return register x0. The ret instruction pops the return address from the stack and jumps back to the calling function.

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An Optimized Approach: From O(N²) to O(N)

The brute-force solution is robust but inefficient. It has a time complexity of roughly O(N²), because of its two nested loops. We can do much better by applying some algebra to eliminate the inner loop entirely.

Let's revisit our two starting equations:

1. a + b + c = N
2. a² + b² = c²

From equation (1), we can express c as c = N - a - b. Now, substitute this into equation (2):

a² + b² = (N - a - b)²

Expand the right side:

a² + b² = N² + a² + b² - 2Na - 2Nb + 2ab

We can cancel out the a² and b² terms from both sides:

0 = N² - 2Na - 2Nb + 2ab

Now, we can isolate the terms with b to solve for it:

2Nb - 2ab = N² - 2Na

Factor out b on the left and N on the right:

b(2N - 2a) = N(N - 2a)

Finally, we get an expression for b in terms of only N and a:

b = (N * (N - 2a)) / (2N - 2a)

This is a game-changer! Now, instead of a nested loop to *search* for b, we can loop only through a and directly *calculate* the corresponding b. If the result of this calculation is an integer, we have a potential candidate for a triplet. This reduces the algorithm's complexity to O(N).

The Optimized Algorithm

1. Start a single loop for a from 1 up to N/3.
2. For each a, calculate the numerator: num = N * (N - 2a).
3. Calculate the denominator: den = 2N - 2a.
4. Check for integer division: if num % den == 0, then we have a valid integer b.
5. Calculate b = num / den.
6. Calculate c = N - a - b.
7. Verify the ordering: a < b < c.
8. If all conditions hold, we have found a triplet. Store it.

Implementing this in assembly is more complex due to the division, but the performance gains for large `N` would be enormous. It would involve using the udiv instruction and checking the remainder.

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Frequently Asked Questions (FAQ)

What is the Arm64 calling convention?

The standard Arm64 procedure call standard (AAPCS64) specifies how functions pass arguments and return values. The first eight integer or pointer arguments are passed in registers x0-x7. Floating-point arguments use registers v0-v7. The return value is placed in x0. Registers x0-x18 are generally available for use, but some (like x19-x29) must be saved by a function if it modifies them.

Why use 64-bit registers (x) instead of 32-bit (w)?

The problem specifies uint64_t for the input `N` and the triplet values, meaning they are 64-bit unsigned integers. Using the 64-bit x registers is necessary to hold these values without truncation. The 32-bit w registers are simply the lower 32 bits of their corresponding x registers and would not be sufficient.

How can I compile and run this Arm64 assembly code?

On a system with an Arm64 architecture (like a Raspberry Pi 4, Apple Silicon Mac, or an AWS Graviton instance) and GCC toolchain, you can assemble and link the code. Save the code as triplets.s. You would then use a C "driver" program to call the function and print the results. Compile them with commands like: gcc -c main.c -o main.o and as triplets.s -o triplets.o, then link them with gcc main.o triplets.o -o triplets_test.

What are some common pitfalls when writing assembly code?

Common errors include incorrect register usage (overwriting a value you still need), violating the calling convention, stack mismanagement (not balancing pushes and pops), off-by-one errors in loops, and incorrect branching logic (e.g., using b.gt when you meant b.ge).

Is assembly language still relevant today?

Absolutely. While not used for general application development, it is indispensable in several key areas: operating system development, embedded systems programming, writing device drivers, compiler design, reverse engineering, and performance-critical computing where every nanosecond counts. For a deeper understanding of computer systems, learning assembly is invaluable. You can explore more advanced concepts in our complete Arm64-assembly guide.

Can this algorithm handle very large numbers for `N`?

The use of 64-bit integers (uint64_t) provides a massive range. However, the intermediate calculations, especially the squaring (a², b², c²), can cause an overflow. For example, if c is close to 2³², then c² will exceed the 2⁶⁴ limit of a 64-bit register. For extremely large `N`, one would need to implement multi-word arithmetic, which significantly increases the complexity of the code.

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Conclusion: The Power of Low-Level Mastery

Solving the Pythagorean triplet problem in Arm64 assembly is a journey from abstract mathematical theory to concrete hardware instructions. We translated a brute-force, nested-loop algorithm into a sequence of register movements, arithmetic operations, and conditional branches. By carefully managing registers and understanding the flow of logic, we built a functional and efficient solution.

More importantly, we saw how a simple algebraic transformation can dramatically optimize the algorithm, reducing its complexity from quadratic to linear time. This highlights a key principle of software engineering: the best performance gains often come from better algorithms, not just clever coding tricks. Mastering challenges like this one from the kodikra.com curriculum builds a solid foundation, empowering you to write faster, more efficient code, no matter what language you use.

Disclaimer: All code and examples are based on the Armv8-A architecture (Arm64). Instructions and register conventions may differ on other architectures. The provided code assumes a 64-bit Linux-like environment and calling convention.

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Published by Kodikra — Your trusted Arm64-assembly learning resource.