“Methods for finding the given number is prime”
Abstract—This document mentions that some of the methods to check a given number is prime or not and calculating the efficiency of each type and comparing the efficiency among all proposed methods. Clearly indicates which method is more efficient in this type of problem.
I. Introduction
A prime number is a number that is divisible by one and itself. There are various methods there to find a given number is prime or not. The only even prime number is 2. All other even numbers can be divided by 2. If the sum of a number's digits is a multiple of 3, that number can be divided by 3. No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5. Zero and 1 are not considered prime numbers. Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime. To prove whether a number is a prime number, first try dividing it by 2 and see if you get a whole number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).
Here is a table of all prime numbers up to 1,000:
2 3 5 7 11 13 17
19 23 29 31 37 41 43
47 53 59 61 67 71 73
79 83 89 97 101 103 107
09 113 127 131 137 139 149
151 157 163 167 173 179 18
191 193 197 199 211 223 227
229 233 239 241 251 257 263
269 271 277 281 283 293 307
311 313 317 331 337 347 349
353 359 367 373 379 383 389
397 401 409 419 421 431 433
439 443 449 457 461 463 467
479 487 491 499 503 509 521
523 541 547 557 563 569 571
577 587 593 599 601 607 613
617 619 631 641 643 647 653
659 661 673 677 683 691 701
709 719 727 733 739 743 751
757 761 769 773 787 797 809
811 821 823 827 829 839 853
857 859 863 877 881 883 887
907 911 919 929 937 941 947
953 967 971 977 983 991 997
In this paper we are discussing one such method such as “mod” method. “mod” is a mathematical method in which it will find a reminder under the division operation. In method 1 Checking each number from 2 to (n-1) numbers. In this method, we check each and every number that comes under 2 to (n-1) whether they are completely divisible or not. If any number is found in between these two then we can conclude that the given number is prime otherwise it is a non-prime number. In method 2 checking half of the given number. In this version, we need to check only half of the number instead of checking entire numbers. Since in the entire numbers checking takes the maximum amount of time that’s why it has been reduced to half. In method 3 Checking a square root of a given number. In this version, we need to check up to the square root of the number instead of checking entire numbers or up to half of that number. In method 4 a function that determines if a number is prime or not. The function must return 1 if prime, 0 if not. Besides, the function must "return" the value of one of the divisors. The best method for checking a given number is prime or non-prime is method 4 that is using even or odd number method because efficiency is high when compared to other methods like square root(method3), a number divided by 2 (method 2) and check the divisibility of given number starting from 2 to n-1. The efficiency and time complexity of all the above-mentioned methods with the help of the algorithm are discussed below.
II. Algorithm development
Check whether the given number is prime or not under different proposed methods. Comparing the efficiency of algorithms among different solution. The different possible ways to find the given number is prime or not is as shown below.
i. Checking each number from 2 to (n-1) numbers.
ii. Checking an (n/2) half of the given number.
iii. Checking the square root of the given number.
iv. Checking a number based on even or an odd number.
III. Algorithm and analysis of the algorithm
The different approaches to find the given number are prime or not are shown in method 1, method 2, method 3 and method 4.
Method 1:
Algorithm
Input: n {it must be positive integer}
Output: prime or not
{
m ßn-1
for kß 2 to m
{
If (n mod k==0) then {not prime, STOP}
else {continue k}
}
{
n is prime number, STOP}
}
Method 2:
Algorithm
Input: n {it must be positive integer}
Output: prime or not
{
m ßn/2
for kß 2 to m
{
If (n mod k==0) then {not prime, STOP}
else {continue k}
}
{n is prime number, STOP}
}
Method 3:
Algorithm
Input: n {it must be positive integer}
Output: prime or not
{
m ßsqrt(n)
for k<- 2 to m
{
If (n mod k==0) then {not prime, STOP}
else {continue k}
}
{
n is prime number, STOP}
}
}
Method 4:
The algorithm to check the even and odd number then find the given number is prime or not.
{
Input: x, div.
output: prime or non-prime.
if (prime2 (x, div)) then {is a prime number, x exit}
else {not prime number. Divisible by, x, div, exit}
}
even (n)
{
return (!(n mod 2));
}
prime2 (n, divisor)
{
Input : i, is_prime;
divisor ß 0;
if (even (n))
{
if (n==2)
divisorß0;
else
divisorß2;
}
else
{
if (n==1)
divisorß0;
else
For iß3 to i<=sqrt(n) ißi+2
{
if (!(n mod i))
divisorßi;
}
}
is_prime ß divisor;
return (!is_prime);
}
IV. Experimentation and profiling
A priori Analysis:
A Priori Analysis is analyzing the algorithm and calculating the computation time for the algorithm is termed as A Priori Analysis. After having made an analytical review of the algorithm, the calculated time complexity for it is of the O(n-2).
to check n is divisible by k. If remains indivisible
for k=2,3,4,5...(n-1)
then n is prime
else non-prime
computing time complexity:
tmax α (n-2)
tmin α 1
tO α (n-2)
tΩ α 1
(n-2) is a polynomial of degree 1.
Method 1
Time complexity
|
frequency
|
|
Best case
|
Worst case
|
|
2
|
1
|
1
|
3
|
1
|
m-1
|
2
|
1
|
m-2
|
0
|
1
|
0
|
1
|
0
|
m-2
|
Table1 Time complexity for the best-case and the worst-case.
Frequency of Testing:
Best case efficiency:
A best-case is one in which a given number is a prime number. Frequency of testing best-case as follows.
tbest α(2*1+3*1+2*1+0*1+1.0)
tbest α7
tbest =C*7*1 (where an α is replaced by some constant “C’)
tbest αβ*1
tbest α1
Worst-case efficiency:
The worst case is one in which an element, not a prime number. In this condition, algorithm efficiency is as shown.
tworst α 1*2+(m-1)*3+(m-2)*2+0*0+(m-2)*1
tworst α 6m-7
tworst α 6(n-1)-7 ( since m=n-1)
tworst α 6n-13 ...........................< 1 >
This algorithm is a linear algorithm of degree 1.
Method 2:
Frequency of Testing:
Worst-case efficiency:
tworst α 6n-13 (Taken from [1])
tworst α 6(n/2)-13
tworst α 3n-13.....................................< 2 >
We have noticed from the above equation. The worst complexity has been reduced to almost half a value when compared to equation number [1].
Method 3:
Frequency of Testing:
Worst-case efficiency:
tworst α 6n-13 (Taken from [1])
tworst α 6 sqrt(n)-13.....................................< 3 >
we have noticed from the above equation The worst complexity has been reduced to almost square root a value when compared to equation number [1] and [2].
Method 4:
Frequency of Testing:
Worst case efficiency:
tworst α 1*1+0*1+1*1+1*1+1*1+1*1+1*1+1*1+1*1+1*1+1*1
+5*(n-3)+2*n+1*1+1*1+1*1
tworst α 7n-2
A Posteriori Analysis:
A Posteriori Analysis is Analysis is done after the execution of the algorithm in the Computer. This Analysis is also called as Profiling.
Figure 1: Comparing the time complexity of different methods.
Where,
continuous line indicates posterior analysis.
dotted line indicates apriori analysis.
V. Summary / conclusion
The below table shows the number of cycles that needs to be executed to find a solution to the problem.
input
|
n
|
(n/2)
|
Squair root(n)
|
10
|
10
|
5
|
4
|
20
|
20
|
10
|
5
|
30
|
30
|
15
|
6
|
40
|
40
|
20
|
7
|
50
|
50
|
25
|
7
|
100
|
100
|
50
|
10
|
500
|
500
|
250
|
24
|
1000
|
1000
|
500
|
33
|
Table2: comparing the efficiency
Figure 2: Comparing the time complexity of different methods.
By comparing the value given in the table we can conclude that (square root(n)) method 3 gives the maximum efficiency when compared to other proposed methods as shown above table 1.
References
[1] Dudley, Underwood (1978), Elementary number theory (2nd ed.), W. H. Freeman and Co., ISBN 978-0-7167-0076-0, p. 10, section 2.
[2] Hardy, Michael; Woodgold, Catherine (2009). "Prime Simplicity". Mathematical Intelligencer 31 (4): 44–52.
[3] C. S. Ogilvy & J. T. Anderson Excursions in Number Theory, pp. 29–35, Dover Publications Inc., 1988 ISBN 0-486-25778-9.
[4] Apostol, Tom M. (1976), Introduction to Analytic Number Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90163-3, Section 1.6, Theorem 1.13.
[5] Derbyshire, John (2003), Prime Obsession, Joseph Henry Press, Washington, DC, ISBN 978-0-309-08549-6, MR 1968857.
