Non-Square Positive Integers not Sum of Square and Prime/Examples/10

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Example of Non-Square Positive Integers not Sum of Square and Prime

$10$ cannot be expressed as the sum of a square and a prime.


Proof

Testing each $m \in \Z_{>0}$ such that $m^2 < 10$ it is established that there is no solution to $10 - m^2 = p$ where $p$ is prime:

\(\displaystyle 10 - 1^2\) \(=\) \(\displaystyle 9\) which is composite: $9 = 3^2$
\(\displaystyle 10 - 2^2\) \(=\) \(\displaystyle 6\) which is composite: $6 = 2 \times 3$
\(\displaystyle 10 - 3^2\) \(=\) \(\displaystyle 1\) $1$ is not Prime

$\blacksquare$