Triple of Triangular Numbers whose Pairwise Sums are Triangular
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Theorem
The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers:
- $66, 105, 105$
Proof
Throughout we use Closed Form for Triangular Numbers, which gives that the $n$th triangular number can be expressed as:
- $T_n = \dfrac {11 \times 12} 2$
We have:
\(\ds 66\) | \(=\) | \(\ds \frac {11 \times 12} 2\) | and so is triangular | |||||||||||
\(\ds 105\) | \(=\) | \(\ds \frac {14 \times 15} 2\) | and so is triangular |
Then:
\(\ds 66 + 105\) | \(=\) | \(\ds 171\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {18 \times 19} 2\) | and so is triangular |
\(\ds 105 + 105\) | \(=\) | \(\ds 210\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {20 \times 21} 2\) | and so is triangular |
\(\ds 66 + 105 + 105\) | \(=\) | \(\ds 210\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {23 \times 24} 2\) | and so is triangular |
$\blacksquare$
Historical Note
The result Triple of Triangular Numbers whose Pairwise Sums are Triangular was reported in Richard K. Guy's Unsolved Problems in Number Theory, 2nd ed. of $1994$ as having been provided by Charles Ashbacher.
Sources
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $66$