Triangular Number whose Square is Triangular

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): $15$
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