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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33$
- but beware a mistake: $85, 86, 87$ is omitted.