## APPENDIX C

## Proof That If an Integer, *P*, Is Not Evenly Divisible by an Integer Less Than the Square Root of *P*, It Is a Prime Number

*Preliminary Proof*

*Given:*

*R* = the square root of *P*.

*Prove:*

If *P* = *H* * *L*, either *H* = *L* = the square root of *P*, or *H* or *L* is < *R* and the other is > *R*.

*Proof*:

If both were > *R*, then their product would be greater then *P*, and if both were less than *R*, their product would be less than *P*.

*Desired Proof*

*Given:*

There are no integers less than *R* (the square root of *P*) that divide evenly into *P*.

*Prove:*

There are no integers greater than *R* that divide evenly into *P*.

*Proof:*

Assume:*H* is an integer > *R* and that *H* divides evenly into.

Then: Define *L*as *L* = *P* / *H*.

*L* is an integer, since *L* = *P* / *H* and *H* divides evenly into *P*