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

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:

\(\ds 10 - 1^2\)

\(=\)

\(\ds 9\)

which is composite: $9 = 3^2$

\(\ds 10 - 2^2\)

\(=\)

\(\ds 6\)

which is composite: $6 = 2 \times 3$

\(\ds 10 - 3^2\)

\(=\)

\(\ds 1\)

$1$ is not Prime

$\blacksquare$