A240129 Triangular numbers that are squares of triangular numbers.

0, 1, 36

Offset

1,3

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).

Links

Table of n, a(n) for n=1..3.

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.

Example

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

Crossrefs

Cf. A000217, A001110.

Sequence in context: A107736 A261121 A139474 * A023929 A010113 A167264

Adjacent sequences: A240126 A240127 A240128 * A240130 A240131 A240132

Keyword

nonn,fini,full,bref

Author

Jonathan Sondow, Apr 02 2014

Status

approved