Sum of squares to 30000?

Probability Level 1

Let $a, b, c, d...$ be positive integers.

Is it possible to find such a set of integers such that $a + b + c + ...$ = $200$ and $a^2 + b^2 + c^2 + ...$ = $30000$ ?

Cannot be determined No Yes

2 solutions

Denton Young
Aug 25, 2019

$173 + 4 + 4 + 4 + 3 + 2 + 1 + 1 +1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 200$

$173^2 + 4^2 + 4^2 + 4^2 + 3^2 + 2^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 +1^2 = 30000$

Alternatively, $173+7+2+18\times 1=200$ and $173^2+7^2+2^2+18\times 1^2=30000$ . I wonder how many solutions there are. I initially tried a theoretical approach but didn't really get anywhere - do you have any insights about when the answer to the general problem is "yes" and when it's "no"?

Chris Lewis - 1 year, 9 months ago

Third solution: {172, 20, 2, 2, 2, 2}

I don't have any insights about the general problem.

Denton Young - 1 year, 9 months ago

Jon Haussmann
Sep 4, 2019

$\begin{aligned} 172 + 20 + 2 + 2 + 2 + 2 &= 200 \\ 172^2 + 20^2 + 2^2 + 2^2 + 2^2 + 2^2 &= 30000 \end{aligned}$

Sum of squares to 30000?

2 solutions

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