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    <p>The algorithm you're following is called the <a href="http://en.wikipedia.org/wiki/Fermat_primality_test" rel="nofollow noreferrer">Fermat primality test</a>. However, there are several problems with your explanation:</p> <ul> <li><p>You say "confirm that gcd(a,p) is &lt; 1". This doesn't make sense as the gcd will never be less than one. What you can check is that gcd(a,p)==1. If it isn't 1, then p is not a prime. This can detect Carmichael numbers, but may have only a small chance to do so.</p></li> <li><p>The way the test is conducted, is that for a certain value of p, you pick several random values of a and check if a ^ (p-1) % p == 1. If one of them is not 1, then p isn't prime. The more values of a you pick, the better your accuracy.</p></li> <li><p>You certainly can't check for <em>all</em> values of x as you say - since there are too many to check.</p></li> <li><p>There is a fast way to perform modular exponentiation, even for large base and exponent. See the <a href="http://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method" rel="nofollow noreferrer">Wikipedia article</a>. You will still need a method to perform multiplication and modulo on large integers.</p></li> <li><p>The Sieve of Eratosthenes is only useful for finding small primes.</p></li> <li><p>This test may incorrectly determine that Carmichael numbers are prime. Other algorithms such as <a href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test" rel="nofollow noreferrer">Rabin-Miller</a> don't have this problem.</p></li> </ul>
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    1. POAssistance finding prime numbers for cryptosystem
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    1. COI don't really see how the gcd test helps to detect carmichael numbers any better than it detects other composite non-carmichael numbers with at least 3 prime factors. It just randomly looks for common factors.
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      1. USJames K Polk
    2. COSince gcd(a,p) != 1 implies a^(p-1) % p != 1, it is completely unnecessary to check for gcd(a,p)==1. You might have to do this test if you are actually looking for Carmichael numbers.
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