Non-Square Positive Integers not Sum of Square and Prime/Examples/34

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

$34$ cannot be expressed as the sum of a square and a prime.

Testing each $m \in \Z_{>0}$ such that $m^2 < 34$ it is established that there is no solution to $34 - m^2 = p$ where $p$ is prime:

$\ds 34 - 1$

$=$

$\ds 33$

which is composite: $33 = 3 \times 11$

$\ds 34 - 4$

$=$

$\ds 30$

which is composite: $30 = 2 \times 3 \times 5$

$\ds 34 - 9$

$=$

$\ds 25$

which is composite: $25 = 5^2$

$\ds 34 - 16$

$=$

$\ds 18$

which is composite: $18 = 2 \times 3^2$

$\ds 34 - 25$

$=$

$\ds 9$

which is composite: $9 = 3^2$

$\blacksquare$