# 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.