| Topic: | Number Theory (Primes) |
|---|---|
| Author: | Peter Bui <pbui@cse.nd.edu> |
| Date: | October 28, 2009 |
Study of the properties and characteristics of numbers, in this case integers.
Lots of possible topics: primes, divisbility (GCD, LCM), modular arithmetic, congruences.
A prime number is an integer p > 1 which is only divisible by 1 and itself:
2 3 5 7 11 13 17
Non-primes are called composite numbers.
Fundamental theorem of arithmetic: Every integer can be expressed in only one way as a product of primes.
Simply try to divide all integers from 2 to sqrt(n) + 1
static int is_prime(int n) {
int i;
if (n == 2)
return TRUE;
for (i = 2; i < sqrt(n) + 1; i++)
if ((n % i) == 0)
return FALSE;
return TRUE;
}
static void prime_factorization(int n) {
int i;
while ((n % 2) == 0)
printf(" %d", 2); n /= 2;
i = 3;
while (i < (sqrt(n) + 1)) {
if ((n % i) == 0)
printf(" %d", i); n /= i;
else
i += 2;
}
if (n > 1) printf(" %d", n);
}
Set all elements in Prime array to TRUE, and then go through composites and mark them as FALSE.
static void compute_primes(int n) {
int i, j;
for (i = 0; i < n; i++)
Prime[i] = TRUE;
for (i = 2; i < n; i++)
for (j = 2; j < n && (i * j) < n; j++)
Prime[i * j] = FALSE;
}