Discussion
Which of the following natural numbers cannot be written as a sum of 3 squares of non-negative integers ?
Answer: B
The number is 2015.
Legendre's three-square theorem: a natural number is a sum of three squares if and only if it is not of the form \(4^{a}(8b + 7)\). None of the options is divisible by 4, so check each modulo 8:
\[ \begin{aligned} 2013 &\equiv 5, \quad 2015 \equiv 7 \\ 2017 &\equiv 1, \quad 2019 \equiv 3 \pmod 8 \end{aligned} \]
Only 2015 is \(\equiv 7 \pmod 8\), so it alone cannot be written as a sum of three squares. (A square is 0, 1 or 4 mod 8, and no three of these add to 7.)
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