Triangular Numbers which are Product of 3 Consecutive Integers
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Theorem
The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are:
- $6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$
This sequence is A001219 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds T_3\) | \(=\) | \(\ds \frac {3 \left({3 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times 2 \times 3\) |
\(\ds T_{15}\) | \(=\) | \(\ds \frac {15 \left({15 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 120\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 5 \times 6\) |
\(\ds T_{20}\) | \(=\) | \(\ds \frac {20 \left({20 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 210\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5 \times 6 \times 7\) |
\(\ds T_{44}\) | \(=\) | \(\ds \frac {44 \left({44 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 990\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 \times 10 \times 11\) |
\(\ds T_{608}\) | \(=\) | \(\ds \frac {608 \left({608 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 185 \, 136\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 56 \times 57 \times 58\) |
\(\ds T_{22 \, 736}\) | \(=\) | \(\ds \frac {22 \, 736 \left({22 \, 736 + 1}\right)} 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 258 \, 474 \, 216\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3 \times 7^2 \times 11 \times 13 \times 29 \times 53\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({2^2 \times 3 \times 53}\right) \times \left({7^2 \times 13}\right) \times \left({2 \times 11 \times 29}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 636 \times 637 \times 638\) |
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Sources
- February 1989: N. Tzanakis and B.M.M. de Weger: On the practical solution of the Thue equation (J. Number Theor. Vol. 31, no. 2: pp. 99 – 132)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $258,474,216$