# Is a Recursive-Iterative Method Better than a Purely Iterative Method to find out if a number is prime?

I made this program in C that tests if a number is prime. I'm as yet unfamiliar with Algorithm complexity and all that Big O stuff, so I'm unsure if my approach, which is a combination of iteration and recursion, is actually more efficient than using a purely iterative method.

#include<stdio.h>
#include<stdlib.h>
#include<math.h>

typedef struct primenode{
long int key;
struct primenode * next;
}primenode;

typedef struct{
primenode * tail;
primenode * curr;
unsigned long int size;
}primelist;

int isPrime(long int number, primelist * list ,long int * calls, long int * searchcalls);
primenode * primelist_insert(long int prime, primelist * list);
int primelist_search(long int searchval, primenode * searchat, long int * calls);
void primelist_destroy(primenode * destroyat);

int main(){
long int n;
long int callstoisprime = 0;
long int callstosearch = 0;
int result = 0;
primelist primes;

//Initialize primelist
primes.tail = NULL;
primes.size = 0;

//Insert 2 as a default prime (optional step)
primelist_insert(2, &primes);

scanf("%d",&n);
printf("Please wait while I crunch the numbers...");
result = isPrime(n, &primes, &callstoisprime, &callstosearch);
switch(result){
case 1: printf("\n%ld is a prime.",n); break;
case -1: printf("\n%ld is a special case. It's neither prime nor composite.",n); break;
default: printf("\n%ld is composite.",n); break;
}
printf("\n\n%d calls made to function: isPrime()",callstoisprime);
printf("\n%d calls made to function: primelist_search()",callstosearch);

//Print all prime numbers in the linked list
printf("\n\nHere are all the prime numbers in the linked list:\n\n");
while(primes.curr != NULL){
printf("%ld ", primes.curr->key);
primes.curr = primes.curr->next;
}
printf("\n\nNote: Only primes up to the square root of your number are listed.\n"
"If your number is negative, only the smallest prime will be listed.\n"
"If your number is a prime, it will itself be listed.\n\n");

//Free up linked list before exiting

return 0;
}

int isPrime(long int number, primelist * list ,long int * calls, long int *searchcalls){
//Returns 1 if prime
//          0 if composite
//          -1 if special case
*calls += 1;
long int i = 2;
if(number==0||number==1){
return -1;
}
if(number<0){
return 0;
}
//Search for it in the linked list of previously found primes
return 1;
}
//Go through all possible prime factors up to its square root
for(i = 2; i <= sqrt(number); i++){
if(isPrime(i, list,calls,searchcalls)){
if(number%i==0) return 0; //It's not a prime
}
}
primelist_insert(number, list); /*Insert into linked list so it doesn't have to keep checking
if this number is prime every time*/
return 1;
}

primenode * primelist_insert(long int prime, primelist * list){
list->curr = malloc(sizeof(primenode));
list->curr->next = NULL;

}
else{
list->tail->next = list->curr;
}
list->tail = list->curr;
list->curr->key = prime;
list->size += 1;

return list->curr;
}

int primelist_search(long int searchval, primenode * searchat, long int * calls){
*calls += 1;
if(searchat == NULL) return 0;
if(searchat->key == searchval) return 1;
return primelist_search(searchval, searchat->next, calls);
}

void primelist_destroy(primenode * destroyat){
if(destroyat == NULL) return;
primelist_destroy(destroyat->next);
free(destroyat);
return;
}


Basically, a lot of what I've seen simple primalty tests do is: 0. 2 is a prime. 1. Cycle through all integers from 2 to half or the square root of the number being tested. 2. If the number is divisible by anything, break and return false; it's composite. 3. Otherwise, return true after the last iteration; it's prime.

I figured that you don't have to test against every number from 2 to the square root, just every prime number, because all other numbers are multiples of primes. So, the function calls itself to find out if a number is prime before using the modulus on it. This works, but I thought it a bit tedious to keep testing all those primes over and over again. So, I used a linked list to store every prime found in it as well, so that before testing primalty, the program searches the list first.

Is it really faster, or more efficient, or did I just waste a lot of time? I did test it on my computer, and for the larger primes it did seem faster, but I'm not sure. I also don't know if it uses significantly more memory since Task Manager just stays a constant 0.7 MB whatever I do.

• Recursion never gives the fastest possible algorithm, because the function calling/return overhead takes far longer to execute than the actual comparison. Also, recursion likely forces the program to use stack where it could have used CPU registers. If you want the fastest possible algorithm, start with unrolling the recursion to a plain loop. Commented Oct 31, 2013 at 10:48