Is Summation a perfect square?

$\sum_{i=1}^{n^2} (i) = a^2$

$(n,a)\in N$ , $N$ denotes natural number.

Does there exist only one such $a$ to satisfy the above conditions?

#NumberTheory

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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}`	$\boxed{123}$

Comments

No, there are actually infinitely many $a$ .
The Pell Equation, $x^2 - 2y^2 = -1$ has infinite solutions, eg, $(1,1), (7,5), (41, 29)$ .
So, we can have, $\sum_{i = 1}^{49} i = \frac{49\cdot 50}{2} = (35)^2$

Ameya Daigavane - 5 years ago

Where $x=n$ and $y=\frac{a}{n}$

A Former Brilliant Member - 5 years ago

Yes, I wanted him to see how the two were related :P

Ameya Daigavane - 5 years ago

@Ameya Daigavane – Oh. Too bad I spoilt it then. :(

A Former Brilliant Member - 5 years ago

Yes I got the above expression, but couldn't find the other one.

Akash Shukla - 5 years ago

Other one?

Ameya Daigavane - 5 years ago

@Ameya Daigavane – As there are infinite $a$ , so I can't find other value of $a$ .

Akash Shukla - 5 years ago

@Akash Shukla – If you look at Deeparaj's comment or after some simple manipulations, you'll see $41 \cdot 29 = 1189$ is another value of $a$ .

Ameya Daigavane - 5 years ago

@Ameya Daigavane – Yes I got this. Thank you so much. it has wonderful connection with pell equation. How do you know that pell equation and my question are related

Akash Shukla - 5 years ago

@Akash Shukla – $\frac{n^2 + 1}{2}$ had to be a perfect square.

Ameya Daigavane - 5 years ago

@Ameya Daigavane – OH!, Yes. You mean $\dfrac{x^2 + 1}{2} = y^2 \implies$ a perfect square

Akash Shukla - 5 years ago

@Akash Shukla – Yes, I changed the variables, as shown in the other comments.

Ameya Daigavane - 5 years 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}`	$\boxed{123}$