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): $15$