Do there exist 10 distinct integers such that sum of any 9 of which is a perfect square?

Do there exist 10 distinct integers such that sum of any 9 of which is a perfect square?

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TopNewestSuppose that the numbers are \(x_1,x_2,\ldots,x_{10}\), and suppose that the sum of all the numbers except the \(j\)th is the square \(y_j^2\). If \(N = x_1+x_2+\cdots+x_{10}\), we deduce that \[ x_j \; = \; N - y_j^2 \qquad 1 \le j \le 10 \] For this to be consistent, we need \[ N \; = \; x_1 + x_2 + \cdots + x_{10} \; = \; 10N - (y_1^2 + y_2^2 + \cdots + y_{10}^2) \] so that \[ y_1^2 + y_2^2 + \cdots + y_{10}^2 \; = \; 9N \] We need to choose \(y_1,y_2,\cdots,y_{10}\) to be distinct integers such that the sum of their squares is divisible by \(9\), for example \(2,3,4,5,6,7,8,9,11,12\). This gives \(N=61\) and \[ x_1,x_2,\ldots,x_{10} \; = \; 57,52,45,36,25,12,-3,-20,-60,-83 \] There is no rule that says that the integers have to be positive. – Mark Hennings · 3 years, 9 months ago

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– Piyushkumar Palan · 3 years, 9 months ago

Wow..thanks a lot.Log in to reply

\(2,3,4,5,6,7,8,9,11,12\)....right? If not,could you please tell me how you found those integers?? – Vaibhav Reddy · 3 years, 9 months agoLog in to reply

– Mark Hennings · 3 years, 9 months ago

Not really. There are lots of options. I could have chosen \(0,9,18,27,\ldots\), but I wanted small numbers.Log in to reply

– Vaibhav Reddy · 3 years, 9 months ago

so you arithmetically solved it on a piece of paperLog in to reply

– Mark Hennings · 3 years, 9 months ago

Well, I played with the values of squares modulo \(9\).Log in to reply

– Vaibhav Reddy · 3 years, 9 months ago

Thank you....I realized it later....Log in to reply

This may be simple or difficult ..i tried for a while...need help! – Piyushkumar Palan · 3 years, 9 months ago

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Why graph of quadratic equation is parabola ? – Vaibhav Gandhapwad · 2 years, 2 months ago

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