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Number Theory Level 3

Find the largest positive integer n n such that 1 2 + 2 2 + 3 2 + … + n 2 {1}^{2}+{2}^{2}+{3}^{2}+\ldots+{n}^{2} is a perfect square.


The answer is 24.

Mehul Arora by Mehul Arora , 20, India

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1 solution

Bob Kadylo
Apr 3, 2017

The only two values of n n for which the indicated series has a Sum which is a perfect square are: 1 and 24.

If n = 1 n=1 the Sum is 1 \boxed{1} and if n = 24 n=24 the Sum is 4900 \boxed{4900} I have checked for 1 ≤ n ≤ 1 , 000 , 000 1\leq n \leq 1,000,000 while killing time in the hospital using just my TI 84+ calculator.

So, the largest value of n n is 24 \boxed{\boxed{24}}

Also, you can check out A000330 on OEIS for further information.

A more general situation with proofs is discussed here .

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