Distinct Primes Factors

Problem Statement

The first two consecutive numbers to have two distinct prime factors are:

• 14 = 2 × 7
• 15 = 3 × 5

The first three consecutive numbers to have three distinct prime factors are:

• 644 = 2² × 7 × 23
• 645 = 3 × 5 × 43
• 646 = 2 × 17 × 19

Find the first four consecutive integers to have four distinct prime factors each. What is the first of these numbers?

Approach

The solution involves:

1. Computing the number of distinct prime factors for each integer
2. Checking consecutive integers for the required property
3. Using efficient prime factorization
4. Finding the first occurrence of four consecutive integers with four distinct prime factors each