Is Summation a perfect square?

i=1n2(i)=a2\sum_{i=1}^{n^2} (i) = a^2

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

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

Note by Akash Shukla
3 years, 4 months ago

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No, there are actually infinitely many a a .
The Pell Equation, x22y2=1 x^2 - 2y^2 = -1 has infinite solutions, eg, (1,1),(7,5),(41,29) (1,1), (7,5), (41, 29) .
So, we can have, i=149i=49502=(35)2 \sum_{i = 1}^{49} i = \frac{49\cdot 50}{2} = (35)^2

Ameya Daigavane - 3 years, 4 months ago

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Where x=nx=n and y=any=\frac{a}{n}

Deeparaj Bhat - 3 years, 4 months ago

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

Ameya Daigavane - 3 years, 4 months ago

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

Deeparaj Bhat - 3 years, 4 months ago

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

Akash Shukla - 3 years, 4 months ago

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Other one?

Ameya Daigavane - 3 years, 4 months ago

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

Akash Shukla - 3 years, 4 months ago

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@Akash Shukla If you look at Deeparaj's comment or after some simple manipulations, you'll see 4129=1189 41 \cdot 29 = 1189 is another value of a a .

Ameya Daigavane - 3 years, 4 months 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 - 3 years, 4 months ago

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@Akash Shukla n2+12\frac{n^2 + 1}{2} had to be a perfect square.

Ameya Daigavane - 3 years, 4 months ago

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@Ameya Daigavane OH!, Yes. You mean x2+12=y2    \dfrac{x^2 + 1}{2} = y^2 \implies a perfect square

Akash Shukla - 3 years, 4 months ago

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

Ameya Daigavane - 3 years, 4 months ago

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