Square Pyramidal and Triangular Numbers

Theorem

The only positive integers which are simultaneously square pyramidal and triangular are:

$1, 55, 91, 208 \, 335$

This sequence is A039596 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

$\ds 1$

$=$

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

Closed Form for Square Pyramidal Numbers

$\ds $

$=$

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

Closed Form for Triangular Numbers

$\ds 55$

$=$

$\ds \dfrac {5 \paren {5 + 1} \paren {2 \times 5 + 1} } 6$

Closed Form for Square Pyramidal Numbers

$\ds $

$=$

$\ds \dfrac {10 \times \paren {10 + 1} } 2$

Closed Form for Triangular Numbers

$\ds 91$

$=$

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

Closed Form for Square Pyramidal Numbers

$\ds $

$=$

$\ds \dfrac {13 \times \paren {13 + 1} } 2$

Closed Form for Triangular Numbers

$\ds 208 \, 335$

$=$

$\ds \dfrac {85 \paren {85 + 1} \paren {2 \times 85 + 1} } 6$

Closed Form for Square Pyramidal Numbers

$\ds $

$=$

$\ds \dfrac {645 \times \paren {645 + 1} } 2$

Closed Form for Triangular Numbers

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Sources

1968: E.T. Avanesov: On a question of a certain theorem of Skolem (Akad. Nauk. Armjan. SSR Vol. 3: pp. 160 – 165)

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $55$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $196,560$
1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $55$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $208,335$