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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
–
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

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## Comments

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TopNewestNo, 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, $\sum_{i = 1}^{49} i = \frac{49\cdot 50}{2} = (35)^2$

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

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

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

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Other one?

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$a$, so I can't find other value of $a$.

As there are infiniteLog in to reply

$41 \cdot 29 = 1189$ is another value of $a$.

If you look at Deeparaj's comment or after some simple manipulations, you'll seeLog in to reply

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

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$\dfrac{x^2 + 1}{2} = y^2 \implies$ a perfect square

OH!, Yes. You meanLog in to reply

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