# Triangular Number whose Square is Triangular

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## Theorem

The only triangular number with less than $660$ digits, whose square is also triangular, is $6$.

## Proof

We have that:

- ${T_3}^2 = 6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$

To establish that it is the only one yet known can be established by an exhaustive search.

$\blacksquare$

## Sources

- 1964: Albert H. Beiler:
*Recreations in the Theory of Numbers* - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $15$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $15$