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

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

Does there exist only one such \(a\) to satisfy the above conditions?

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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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