Non-Square Positive Integers not Sum of Square and Prime/Examples/58

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

$58$ cannot be expressed as the sum of a square and a prime.

Testing each $m \in \Z_{>0}$ such that $m^2 < 58$ it is established that there is no solution to $58 - m^2 = p$ where $p$ is prime:

$\ds 58 - 1$

$=$

$\ds 57$

which is composite: $57 = 3 \times 19$

$\ds 58 - 4$

$=$

$\ds 54$

which is composite: $54 = 2 \times 3^3$

$\ds 58 - 9$

$=$

$\ds 49$

which is composite: $49 = 7^2$

$\ds 58 - 16$

$=$

$\ds 42$

which is composite: $42 = 2 \times 3 \times 7$

$\ds 58 - 25$

$=$

$\ds 33$

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

$\ds 58 - 36$

$=$

$\ds 22$

which is composite: $22 = 2 \times 11$

$\ds 58 - 49$

$=$

$\ds 9$

which is composite: $9 = 3^2$

$\blacksquare$