Triplets of Products of Two Distinct Primes

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Theorem

The following triplets of consecutive positive integers are the smallest in which each number is the product of $2$ distinct prime numbers:

$33, 34, 35$
$85, 86, 87$
$93, 94, 95$
$141, 142, 143$
$201, 202, 203$
$213, 214, 215$
$217, 218, 219$


Proof

Taking each triplet in turn:


\(\ds 33\) \(=\) \(\ds 3 \times 11\)
\(\ds 34\) \(=\) \(\ds 2 \times 17\)
\(\ds 35\) \(=\) \(\ds 5 \times 7\)


\(\ds 85\) \(=\) \(\ds 5 \times 17\)
\(\ds 86\) \(=\) \(\ds 2 \times 43\)
\(\ds 87\) \(=\) \(\ds 3 \times 29\)


\(\ds 93\) \(=\) \(\ds 3 \times 31\)
\(\ds 94\) \(=\) \(\ds 2 \times 47\)
\(\ds 95\) \(=\) \(\ds 5 \times 19\)


\(\ds 141\) \(=\) \(\ds 3 \times 47\)
\(\ds 142\) \(=\) \(\ds 2 \times 71\)
\(\ds 143\) \(=\) \(\ds 11 \times 13\)


\(\ds 201\) \(=\) \(\ds 3 \times 67\)
\(\ds 202\) \(=\) \(\ds 2 \times 101\)
\(\ds 203\) \(=\) \(\ds 7 \times 29\)


\(\ds 213\) \(=\) \(\ds 3 \times 71\)
\(\ds 214\) \(=\) \(\ds 2 \times 107\)
\(\ds 215\) \(=\) \(\ds 5 \times 43\)


\(\ds 217\) \(=\) \(\ds 7 \times 31\)
\(\ds 218\) \(=\) \(\ds 2 \times 109\)
\(\ds 219\) \(=\) \(\ds 3 \times 73\)


It is noted that the triplet:

$121, 122, 123$

while consisting of semiprimes, not all of these are the product of $2$ distinct prime numbers, as $121 = 11^2$.

$\blacksquare$


Sources

but beware a mistake: $85, 86, 87$ is omitted.