Product of Two Triangular Numbers to make Square/Examples/T3 by T48

Example of Product of Two Triangular Numbers to make Square

$T_3 \times T_{48} = 84^2$

where $T_3, T_{48}$ denote the $2$nd and $48$th triangular numbers.

Proof

\(\ds T_3 \times T_{48}\)

\(=\)

\(\ds \dfrac {3 \paren {3 + 1} } 2 \times \dfrac {48 \paren {48 + 1} } 2\)

Closed Form for Triangular Numbers

\(\ds \)

\(=\)

\(\ds 3 \times 2 \times 24 \times 49\)

\(\ds \)

\(=\)

\(\ds \paren {3 \times 2} \times \paren {3 \times 2^3} \times 7^2\)

\(\ds \)

\(=\)

\(\ds \paren {3 \times 2^2 \times 7}^2\)

\(\ds \)

\(=\)

\(\ds 84^2\)

$\blacksquare$