# Arithmetic Progression of 4 Terms with 3 Distinct Prime Factors

## Theorem

$30, 66, 102, 138$

is the smallesr of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors.

## Proof

We demonstrate that this is indeed an arithmetic progression:

 $\displaystyle 66 - 30$ $=$ $\displaystyle 36$ $\displaystyle 102 - 66$ $=$ $\displaystyle 36$ $\displaystyle 138 - 102$ $=$ $\displaystyle 36$

demonstrating the common difference of $36$.

Then we note:

 $\displaystyle 30$ $=$ $\displaystyle 2 \times 3 \times 5$ $\displaystyle 66$ $=$ $\displaystyle 2 \times 3 \times 11$ $\displaystyle 102$ $=$ $\displaystyle 2 \times 3 \times 17$ $\displaystyle 138$ $=$ $\displaystyle 2 \times 3 \times 23$

It remains to be shown that it is the smallest such arithmetic progression.