Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors

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Theorem

The arithmetic sequence:

$30, 66, 102, 138$

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


Proof

We demonstrate that this is indeed an arithmetic sequence:

\(\ds 66 - 30\) \(=\) \(\ds 36\)
\(\ds 102 - 66\) \(=\) \(\ds 36\)
\(\ds 138 - 102\) \(=\) \(\ds 36\)

demonstrating the common difference of $36$.


Then we note:

\(\ds 30\) \(=\) \(\ds 2 \times 3 \times 5\)
\(\ds 66\) \(=\) \(\ds 2 \times 3 \times 11\)
\(\ds 102\) \(=\) \(\ds 2 \times 3 \times 17\)
\(\ds 138\) \(=\) \(\ds 2 \times 3 \times 23\)


It is understood that smallest arithmetic sequence means:

integer sequence of length $4$ whose terms are in arithmetic sequence such that their sum is the smallest of all such sequences.


To demonstrate that this sequence is indeed the smallest, according to this definition, we list all positive integers less than $138$, each with $3$ distinct prime factors.

They are:

\(\ds 30\) \(=\) \(\ds 2 \times 3 \times 5\)
\(\ds 42\) \(=\) \(\ds 2 \times 3 \times 7\)
\(\ds 66\) \(=\) \(\ds 2 \times 3 \times 11\)
\(\ds 70\) \(=\) \(\ds 2 \times 5 \times 7\)
\(\ds 78\) \(=\) \(\ds 2 \times 3 \times 13\)
\(\ds 102\) \(=\) \(\ds 2 \times 3 \times 17\)
\(\ds 105\) \(=\) \(\ds 3 \times 5 \times 7\)
\(\ds 110\) \(=\) \(\ds 2 \times 5 \times 11\)
\(\ds 114\) \(=\) \(\ds 2 \times 3 \times 17\)
\(\ds 130\) \(=\) \(\ds 2 \times 5 \times 13\)

One cannot pick any other $4$ numbers from the list to form an arithmetic sequence.

Hence any further such arithmetic sequence will contain larger terms whose sum will be consequently larger.

$\blacksquare$


Sources

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