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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?

Note by Akash Shukla
5 months, 3 weeks ago

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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 · 5 months, 3 weeks ago

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@Ameya Daigavane Yes I got the above expression, but couldn't find the other one. Akash Shukla · 5 months, 3 weeks ago

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@Akash Shukla Other one? Ameya Daigavane · 5 months, 3 weeks ago

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@Ameya Daigavane As there are infinite \(a\), so I can't find other value of \(a\). Akash Shukla · 5 months, 3 weeks ago

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@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 months, 3 weeks ago

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@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 months, 3 weeks ago

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@Akash Shukla \(\frac{n^2 + 1}{2} \) had to be a perfect square. Ameya Daigavane · 5 months, 3 weeks ago

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@Ameya Daigavane OH!, Yes. You mean \(\dfrac{x^2 + 1}{2} = y^2 \implies\) a perfect square Akash Shukla · 5 months, 3 weeks ago

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@Akash Shukla Yes, I changed the variables, as shown in the other comments. Ameya Daigavane · 5 months, 3 weeks ago

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@Ameya Daigavane Where \(x=n\) and \(y=\frac{a}{n} \) Deeparaj Bhat · 5 months, 3 weeks ago

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@Deeparaj Bhat Yes, I wanted him to see how the two were related :P Ameya Daigavane · 5 months, 3 weeks ago

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@Ameya Daigavane Oh. Too bad I spoilt it then. :( Deeparaj Bhat · 5 months, 3 weeks ago

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