Arithmetic Progression of 4 Terms with 3 Distinct Prime Factors

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Theorem

The arithmetic progression:

$30, 66, 102, 138$

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:

\(\displaystyle 66 - 30\) \(=\) \(\displaystyle 36\) $\quad$ $\quad$
\(\displaystyle 102 - 66\) \(=\) \(\displaystyle 36\) $\quad$ $\quad$
\(\displaystyle 138 - 102\) \(=\) \(\displaystyle 36\) $\quad$ $\quad$

demonstrating the common difference of $36$.


Then we note:

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


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



Sources