Watch 9 video solutions for Prime Pairs With Target Sum, a medium level problem involving Array, Math, Enumeration. This walkthrough by Tech Courses has 947 views views. Want to try solving it yourself? Practice on FleetCode or read the detailed text solution.
You are given an integer n. We say that two integers x and y form a prime number pair if:
1 <= x <= y <= nx + y == nx and y are prime numbersReturn 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.
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 <= 106Problem Overview: Given an integer n, return all pairs of prime numbers [x, y] such that x + y = n and x ≤ y. The task combines prime number generation with efficient pair enumeration.
Approach 1: Trial Division with Reduction (Time: O(n√n), Space: O(1))
Iterate the first value a from 2 to n / 2. For each candidate, compute b = n - a. Check if both numbers are prime using trial division up to √x. If both checks pass, record the pair [a, b]. This approach uses pure math and number theory logic without extra memory, but repeated primality checks make it slower as n grows.
Approach 2: Sieve of Eratosthenes with Pair Search (Time: O(n log log n) + O(n), Space: O(n))
Precompute all primes up to n using the Sieve of Eratosthenes. The sieve builds a boolean array where isPrime[i] indicates whether i is prime. After preprocessing, iterate a from 2 to n / 2 and check if both a and n - a are marked prime. Each lookup is constant time, so the pair search becomes a simple linear scan. This approach leverages efficient prime generation from number theory and straightforward array lookups, making it the most practical solution for larger inputs.
Recommended for interviews: Start by explaining the trial division approach because it shows you understand primality testing and the pair constraint a + b = n. Then optimize with the Sieve of Eratosthenes. Interviewers typically expect the sieve-based solution since it reduces repeated prime checks and achieves near-linear preprocessing time with constant-time lookups.
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Trial Division with Reduction | O(n√n) | O(1) | When memory is constrained or when explaining the basic mathematical idea first |
| Sieve of Eratosthenes with Pair Search | O(n log log n) + O(n) | O(n) | Best general solution when n can be large and many prime checks are required |