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Question Number 165098 by mnjuly1970 last updated on 26/Jan/22

	Answered by Eulerian last updated on 26/Jan/22

	$Solution :$ $We know that :$ $1^{2} + 2^{2} + 3^{2} + . . . . . . + n^{2} = \frac{n (n + 1) (2 n + 1)}{6}$ $∴$ $\frac{n (n + 1) (2 n + 1)}{6} = m^{2}$ ${6 m}^{2} = (n^{2} + n) (2 n + 1)$ ${6 m}^{2} = {2 n}^{3} + {3 n}^{2} + n$ ${3 m}^{2} = n^{3} + \frac{{3 n}^{2}}{2} + \frac{n}{2}$ $One can easily notice that when (m, n) = (1, 1)$ $the equation satisfies the condition . By transforming$ $the right hand side into a depressed cubic polynomial, let :$ $n = z - \frac{(\frac{3}{2})}{3} = z - \frac{1}{2}$ $∴$ ${(z - \frac{1}{2})}^{3} + \frac{3}{2} \cdot {(z - \frac{1}{2})}^{2} + \frac{1}{2} \cdot (z - \frac{1}{2}) = {3 m}^{2}$ $z^{3} + (\frac{1}{2} - \frac{{(\frac{3}{2})}^{2}}{3}) \cdot z + \frac{2 {(\frac{3}{2})}^{3} - 9 (\frac{3}{2}) (\frac{1}{2})}{27} = {3 m}^{2}$ $z^{3} - \frac{z}{4} = {3 m}^{2}$ $let :$ ${3 m}^{2} = y^{2}$ $Notice that we now have a form of elliptic curve equation :$ $y^{2} = z^{3} - \frac{z}{4}$ $And from our previous solution, it works at (y, z) = (\pm \sqrt{3}, \frac{3}{2}) .$ $By building a tangent line to the elliptic curve, we have :$ $y^{'} = \frac{d}{dz} (\sqrt{z^{3} - \frac{z}{4}}) = (\frac{{12 z}^{2} - 1}{{8 z}^{3} - 2 z}) \cdot \sqrt{z^{3} - \frac{z}{4}}$ $m_{\tan} = \frac{13 \sqrt{3}}{12}$ $∴$ $y - y_{1} = m_{\tan} (z - z_{1})$ $y = \frac{13 \sqrt{3}}{12} \cdot (z - \frac{3}{2}) + \sqrt{3}$ $Thus, we can now see all of the possible solutions to the equation :$ $\frac{13 \sqrt{3}}{12} \cdot (z - \frac{3}{2}) + \sqrt{3} = \sqrt{z^{3} - \frac{z}{4}}$ $z = \frac{25}{48}$ $Therefore, we now have :$ $y^{2} = {(\frac{25}{48})}^{3} - (\frac{1}{4}) (\frac{25}{48})$ $y = \pm \frac{35 \sqrt{3}}{576}$ $∴$ $(y, z) = (\pm \frac{35 \sqrt{3}}{576}, \frac{25}{48}), (\pm \sqrt{3}, \frac{3}{2})$ $By substituting back, we now have :$ ${3 m}^{2} = {(\pm \frac{35 \sqrt{3}}{576})}^{2}$ $m = \pm \frac{35}{576}$ $and$ $\frac{25}{48} - \frac{1}{2} = n$ $n = \frac{1}{48}$ $∴$ $(m, n) = (\pm 1, 1), (\pm \frac{35}{576}, \frac{1}{48})$ $By graphing the elliptic curve and its tangent line :$
	Commented by Rasheed.Sindhi last updated on 26/Jan/22

	$n, m m u s t b e n a t u r a l!$
	Answered by Sheenaynay last updated on 26/Jan/22

	$\frac{n \times (n + 1) \times (2 n + 1)}{6} = m^{2}$ $n (n + 1) (2 n + 1) = m^{2} \times 6$ $n = 24 \land m = 70$
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