Index of Square Triangular Number from Preceding
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Theorem
Let $T_n$ be the $n$th triangular number.
Let $T_n$ be square.
Then $T_{4 n \paren {n + 1} }$ is also square.
Proof
\(\ds T_{4 n \paren {n + 1} }\) | \(=\) | \(\ds \frac {\paren {4 n \paren {n + 1} } \paren {4 n \paren {n + 1} + 1} } 2\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds T_{4 n \paren {n + 1} }\) | \(=\) | \(\ds \frac {n \paren {n + 1} } 2 \times 4 \paren {4 n \paren {n + 1} + 1}\) | Extract $T_n$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds T_{4 n \paren {n + 1} }\) | \(=\) | \(\ds T_n \times \paren {16 n^2 + 16 n + 4}\) | Simplify | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds T_{4 n \paren {n + 1} }\) | \(=\) | \(\ds T_n \times \paren {4 n + 2}^2\) | Factorise |
The product of two squares is also square.
Let $T_n$ be square.
Therefore $T_{4 n \paren {n + 1} }$ is also square.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$