Square and Tetrahedral Numbers

Theorem

The only positive integers which are simultaneously tetrahedral and square are:

$1, 4, 19 \, 600$

Proof

$\ds 1$

$=$

$\ds \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds 1^2$

Definition of Square Number

$\ds 4$

$=$

$\ds \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds 2^2$

Definition of Square Number

$\ds 19 \, 600$

$=$

$\ds \dfrac {48 \paren {48 + 1} \paren {48 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds 140^2$

Definition of Square Number

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Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $19,600$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19,600$