Square Triangles

A triangle number is one of the form $\sum_{k = 1}^n k = \frac{n(n + 1)}{2}$ for positive any integer n. So my question is how many triangle numbers are also square numbers; that is how many ordered pairs of positive integers $(n,m)$ are there such that $f(n,m) = \frac{n(n + 1)}{2} - m^{2} = 0$ . $(8,6),(49,35)$ are two solutions of $f(n,m) = 0$ ..

Easy Math Editor

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Comments

$\begin{aligned} \frac{n(n+1)}{2} &= m^2 \\ n^2 + n &= 2m^2 \\ 4n^2 + n + 1 = 8m^2 + 1 \end{aligned}$

Then substitute $x=2n+1$ and $y=2m$ :

$\begin{aligned} x^2 &= 2y^2 + 1 \\ x^2 - 2y^2 &= 1 \end{aligned}$

This is a well-known Pell's equation, and has infinite solutions.

Tim Vermeulen - 7 years, 10 months ago

Ok so $g(x,y) = x^{2} - 2y^{2} - 1 = 0$ has infinitely many integral solutions. But each $(x,y)$ that solves $g(x,y) = 0$ gives a $(n,m)$ that solves $f(n,m) = 4n^{2} + 4n - 8m^{2} = 0$ only if $x = 2k + 1$ and $x \ge 3$ .

Samuel Queen - 7 years, 10 months ago

Yeah, and $x^2-2y^2=1$ also has infinite solutions for $x$ being an odd integer. Should have added that.

Tim Vermeulen - 7 years, 10 months ago

@Tim Vermeulen – Isn't $x$ automatically odd? (That is, if $x,y$ are both integers.)

Mitya Boyarchenko - 7 years, 10 months ago

The 3rd equation must be $4n^{2}+4n+1=8m^{2}+1.$

Kishan k - 7 years, 10 months ago

See problem 5 here for another proof that there are infinitely many triangular numbers that are perfect squares.

Jimmy Kariznov - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$