A240129 Triangular numbers that are squares of triangular numbers.

0, 1, 36

Offset: 1

Author

Jonathan Sondow, Apr 02 2014

Keywords

• nonn
• fini
• full
• bref

Comments

Ljunggren used p-adic methods to prove that 0, 1, and 36 are the only triangular numbers that are squares of triangular numbers.

Same as: the only positive integer solutions of (x(x-1))^2 = 2y(y-1) are (x,y) = (1,1), (2,2), and (4,9).

Cassels used elliptic curves to simplify Ljunggren's proof.

Subsequence of A001110 (triangular numbers that are squares).

Examples

			1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 = 6^2 = (1 + 2 + 3)^2.
		

Links

• W. Ljunggren, Review of "Integral points on certain elliptic curves" by J.W.S. Cassels, Proc. Lond. Math. Soc., III. Ser. 14 A (1965), 55-57, zbMATH 0134.27501.

Crossrefs

Cf. A000217, A001110.