Download Prime Number Identifier - Wilfrid Laurier University Physics Labs

Prime Number Identifier
Terry Sturtevant
Department of Physics and Computer Science, Wilfrid Laurier University
Introduction
Simplifying Equation
Finding prime numbers is a common task in mathematics. This circuit will identify the prime
numbers between zero and fifteen.
In this case, a Karnaugh map was used to determine simplified sum-of-products logic equations.
Definition
a3 a2
A number is defined as prime if it has exactly two divisors, itself and one. By definition, the
numbers zero and one are neither prime nor composite.
00
01
11
10
a1 a0
00 01 11 10
0 0 1 1
0 1 1 0
0 1 0 0
0 0 1 0
Circuit Drawing and
Simulation
The circuit looks like this:
Another grouping can be used to get the remaining ones.
Logic Design
For this particular problem, the truth table looks like this:
number binary (a3a2a1a0) p/c/n
0
0000
n
1
0001
n
2
0010
p
3
0011
p
4
0100
c
5
0101
p
6
0110
c
7
0111
p
8
1000
c
9
1001
c
10
1010
c
11
1011
p
12
1100
c
13
1101
p
14
1110
c
15
1111
c
a3 a2
00
01
11
10
a1a0
00 01 11 10
0 0 1 1
0 1 1 0
0 1 0 0
0 0 1 0
The resulting SOP equation is
prime = a3 a2 a1 + a3 a2 a0 + a2 a1 a0 + a2 a1 a0
The simulation output looks like this:
Testing Equation
Maxima was used to test the equation.
2 3
a3 a2
00
01
11
10
a1 a0
00 01 11 10
0 0 1 1
0 1 1 0
0 1 0 0
0 0 1 0
5
7
11
13
You can see that the ouput is only high for the highlighted cases; i.e. where the input number is
prime. This verifies that circuit correctly implements the equation.
The prime numbers are highlighted in red. There are 6 input combinations that give a TRUE
output; for all other input combinations the output is FALSE.
A truth table for the output, prime, which is TRUE for a prime number, looks like the following.
(The inputs have been grey-coded to produce a Karnaugh map.)
The two AND gates which implement the terms highlighted in the Karnaugh can be identified by
coloured dots on them.
2
3
5
7
11
13
Testing
Note that there are exactly 6 cells in the Karnaugh map with a one, corresponding to the six
numbers which are prime. All the other cells are zero.
All possibilities were tested to see that prime was only true for 2,3,5,7,11, and 13. This verifies
that the equation is correct.
Prime should only be true for 2,3,5,7,11, and 13
PC/CP120 Lab 2013