Product of Two Triangular Numbers to make Square/Examples/T2 by T24

Example of Product of Two Triangular Numbers to make Square

$T_2 \times T_{24} = 30^2$

where $T_2, T_{24}$ denote the $2$nd and $24$th triangular numbers.

Proof

\(\ds T_2 \times T_{24}\)

\(=\)

\(\ds \dfrac {2 \paren {2 + 1} } 2 \times \dfrac {24 \paren {24 + 1} } 2\)

Closed Form for Triangular Numbers

\(\ds \)

\(=\)

\(\ds 1 \times 3 \times 12 \times 25\)

\(\ds \)

\(=\)

\(\ds 3 \times \paren {3 \times 2^2} \times 5^2\)

\(\ds \)

\(=\)

\(\ds \paren {3 \times 2 \times 5}^2\)

\(\ds \)

\(=\)

\(\ds 30^2\)

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$

although the example chosen, $T_3 \times T_{24}$, is erroneous.