"Is a number prime?"

Question

I've come up with a very efficient way of finding if a number is prime, using TI-BASIC for use with TI-83/84/+/SE calculators. I am trying to optimize it however possible.

:Input "NUMBER: ",A
:If A<2 or fPart(A
:Then
:Disp "INVALID INPUT"
:Stop
:End
:0→B
:2→I
:While I<A and not(B
:If 0=fPart(A/I
:1→B
:I+2→I
:If I=4
:3→I
:End
:If B
:Disp "NOT"
:Disp "PRIME"

A few notes:

• Lines 2-6 are for validating input, and don't affect the effectiveness of the program.
• The closing "s on line 4 and the last two lines are not necessary, but I added them so the syntax would highlight nicely here.
• Lines 12-14 are for speeding up the loop doubly; instead of incrementing by 1 each time, it is by 2, with the If to offset the 4 to 3 on the first run.
• The not(B was very efficient, to end the loop whenever a match to identify the number as a composite number is found.

Answer 1

I do not speak TI-BASIC, so I cannot review the style, coding conventions etc. But there are two possible optimizations:

• If a number A is composite then it must have a factor that is less than or equal to √(A). So you can replace

    :While I<A and not(B
  

with:

    :√(A)→S
    :While I≤S and not(B

This reduces the number of trial divisions substantially if the input is a prime number.

• Check the divisibility by 2 first, and then loop just over the odd numbers I = 3, 5, 7, .... This saves you from checking

    :If I=4
    :3→I
  

in each loop iteration.