Tetrahedral and Triangular Numbers

Theorem

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

$1, 10, 120, 1540, 7140$

Proof

$\ds 1$

$=$

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

Closed Form for Tetrahedral Numbers

$\ds $

$=$

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

Closed Form for Triangular Numbers

$\ds 10$

$=$

$\ds \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds \dfrac {4 \times \paren {4 + 1} } 2$

Closed Form for Triangular Numbers

$\ds 120$

$=$

$\ds \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds \dfrac {15 \times \paren {15 + 1} } 2$

Closed Form for Triangular Numbers

$\ds 1540$

$=$

$\ds \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds \dfrac {55 \times \paren {55 + 1} } 2$

Closed Form for Triangular Numbers

$\ds 7140$

$=$

$\ds \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6$

Closed Form for Tetrahedral Numbers

$\ds $

$=$

$\ds \dfrac {119 \times \paren {119 + 1} } 2$

Closed Form for Triangular Numbers

This theorem requires a proof.
In particular: It remains to be shown that these are the only such instances.
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Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1540$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1540$