Question 10

What is the number of integers in the set $$\{0, \ldots, 20\}$$ which can be expressed as the sum of two square integers?

Correct Answer: 13

Solution

Only the squares $$0, 1, 4, 9, 16$$ can occur, and negative integers give the same squares. Listing each unordered pair once with the smaller square first and keeping sums at most 20 gives $$0, 1, 4, 9, 16$$ from 0, then $$2, 5, 10, 17$$ from 1, then $$8, 13, 20$$ from 4, and 18 from 9. All these sums are distinct, so the count is $$5 + 4 + 3 + 1 = 13$$.

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