LeetCode #2761 — MEDIUM

Prime Pairs With Target Sum

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

You are given an integer n. We say that two integers x and y form a prime number pair if:

1 <= x <= y <= n
x + y == n
x and y are prime numbers

Return the 2D sorted list of prime number pairs [x_i, y_i]. The list should be sorted in increasing order of x_i. If there are no prime number pairs at all, return an empty array.

Note: A prime number is a natural number greater than 1 with only two factors, itself and 1.

Example 1:

Input: n = 10
Output: [[3,7],[5,5]]
Explanation: In this example, there are two prime pairs that satisfy the criteria. 
These pairs are [3,7] and [5,5], and we return them in the sorted order as described in the problem statement.

Example 2:

Input: n = 2
Output: []
Explanation: We can show that there is no prime number pair that gives a sum of 2, so we return an empty array.

Constraints:

1 <= n <= 10⁶

Step 01

Brute Force Baseline

Problem summary: You are given an integer n. We say that two integers x and y form a prime number pair if: 1 <= x <= y <= n x + y == n x and y are prime numbers Return the 2D sorted list of prime number pairs [xi, yi]. The list should be sorted in increasing order of xi. If there are no prime number pairs at all, return an empty array. Note: A prime number is a natural number greater than 1 with only two factors, itself and 1.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Math

Example 1

Example 2

Core Insight

What unlocks the optimal approach

Pre-compute all the prime numbers in the range [1, n] using a sieve, and store them in a data structure where they can be accessed in O(1) time.
For x in the range [2, n/2], we can use the pre-computed list of prime numbers to check if both x and n - x are primes. If they are, we add them to the result.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
class Solution {
    public List<List<Integer>> findPrimePairs(int n) {
        boolean[] primes = new boolean[n];
        Arrays.fill(primes, true);
        for (int i = 2; i < n; ++i) {
            if (primes[i]) {
                for (int j = i + i; j < n; j += i) {
                    primes[j] = false;
                }
            }
        }
        List<List<Integer>> ans = new ArrayList<>();
        for (int x = 2; x <= n / 2; ++x) {
            int y = n - x;
            if (primes[x] && primes[y]) {
                ans.add(List.of(x, y));
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
func findPrimePairs(n int) (ans [][]int) {
	primes := make([]bool, n)
	for i := range primes {
		primes[i] = true
	}
	for i := 2; i < n; i++ {
		if primes[i] {
			for j := i + i; j < n; j += i {
				primes[j] = false
			}
		}
	}
	for x := 2; x <= n/2; x++ {
		y := n - x
		if primes[x] && primes[y] {
			ans = append(ans, []int{x, y})
		}
	}
	return
}

# Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
class Solution:
    def findPrimePairs(self, n: int) -> List[List[int]]:
        primes = [True] * n
        for i in range(2, n):
            if primes[i]:
                for j in range(i + i, n, i):
                    primes[j] = False
        ans = []
        for x in range(2, n // 2 + 1):
            y = n - x
            if primes[x] and primes[y]:
                ans.append([x, y])
        return ans

// Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
// Rust example auto-generated from java reference.
// Replace the signature and local types with the exact LeetCode harness for this problem.
impl Solution {
    pub fn rust_example() {
        // Port the logic from the reference block below.
    }
}

// Reference (java):
// // Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
// class Solution {
//     public List<List<Integer>> findPrimePairs(int n) {
//         boolean[] primes = new boolean[n];
//         Arrays.fill(primes, true);
//         for (int i = 2; i < n; ++i) {
//             if (primes[i]) {
//                 for (int j = i + i; j < n; j += i) {
//                     primes[j] = false;
//                 }
//             }
//         }
//         List<List<Integer>> ans = new ArrayList<>();
//         for (int x = 2; x <= n / 2; ++x) {
//             int y = n - x;
//             if (primes[x] && primes[y]) {
//                 ans.add(List.of(x, y));
//             }
//         }
//         return ans;
//     }
// }

// Accepted solution for LeetCode #2761: Prime Pairs With Target Sum
function findPrimePairs(n: number): number[][] {
    const primes: boolean[] = new Array(n).fill(true);
    for (let i = 2; i < n; ++i) {
        if (primes[i]) {
            for (let j = i + i; j < n; j += i) {
                primes[j] = false;
            }
        }
    }
    const ans: number[][] = [];
    for (let x = 2; x <= n / 2; ++x) {
        const y = n - x;
        if (primes[x] && primes[y]) {
            ans.push([x, y]);
        }
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n log log n)

Space

O(n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.