Distinct Primes Factors

The first two consecutive numbers to have two distinct prime factors are:
\begin{align}
14 &= 2 \times 7\\
15 &= 3 \times 5.
\end{align}
The first three consecutive numbers to have three distinct prime factors are:
\begin{align}
644 &= 2^2 \times 7 \times 23\\
645 &= 3 \times 5 \times 43\\
646 &= 2 \times 17 \times 19.
\end{align}
Find the first four consecutive integers to have four distinct prime factors each. What is the first of these numbers?

To find the first four consecutive integers that have four distinct prime factors, we have to do a bit of manual calculation. We’re essentially looking for four consecutive integers that can be factored into exactly four prime numbers each.

First, we start by calculating the prime factors for different numbers and note any patterns that emerge. After doing this for a while, you will notice a pattern:

\[
134043 = 3 \times 7 \times 13 \times 491
\]
\[
134044 = 2^2 \times 23 \times 31 \times 47
\]
\[
134045 = 5 \times 17 \times 19 \times 83
\]
\[
134046 = 2 \times 3 \times 59 \times 677
\]

The first number in this sequence is 134043, so this is the first number of the four consecutive integers we were originally looking for.

Keep in mind that this is a computationally intensive problem, and a computer would be optimal for a speedy discovery of the solution. This procedure would take a long time if done manually, even if you were to use a sieving method.

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