What is the smallest member of the set?

Number Theory Level pending

${\color{#D61F06}\text{This problem's question:}}$ What is the smallest member of the set of pairs of non-negative integers which when the integers are squared and then the two squares are summed equals $9876543210$ ?

As the question is not the clearest exposition, here is a small example of the desired process:

If the target value were 100 instead, then the set of pairs of non-negative integers would be $\left( \begin{array}{cc} 0 & 10 \\ 6 & 8 \\ \end{array} \right)$ , reading the pairs row-wise in the matrix and the matrix as a whole being the set of pairs. The desired answer is the smallest integer in the matrix, which in this case is $0$ .

The answer is 3777.

1 solution

A Former Brilliant Member
Jul 25, 2019

Done as a computer search, the answer is as given: 3777.

$n=9876543210;\,l=\left\lfloor \sqrt{n}\right\rfloor ;\, \\ \min \left(\text{Flatten}\left[\text{Union}\left[\text{Table}\left[\text{If}\left[\sqrt{n-i^2}\in \mathbb{Z},\text{Sort}\left[\left\{\sqrt{n-i^2},i\right\}\right],\text{Nothing}\right],\{i,0,l\}\right]\right]\right]\right) \Rightarrow 3777$

The matrix of pairs: $\left( \begin{array}{rr} 3777 & 99309 \\ 18921 & 97563 \\ 29217 & 94989 \\ 43401 & 89403 \\ 58461 & 80367 \\ 62607 & 77181 \\ \end{array} \right)$

If one does the search starting from $0$ , then the first pair found would be the pair with the desired answer.

What is the smallest member of the set?

The answer is 3777.

1 solution

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