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How do I mathematically determine if a number is prime? If the number is n, then dividing it by every prime number less than or equal to sqrt(n) and showing that there is a remainder. There are a number of different sieve solutions for finding prime numbers, the oldest and most famous of which is the Sieve of Eratosthenes. These are generally ...
Is $31!+1$ a prime number? Prime numbers can be written as $6n+1$ or $6n-1$ form. $31!+1$ can also be written as $6n+1$,but not every number of the form $6n+1$ is prime. How do I proceed efficie...
For small numbers trial division is best. Compute the square root of the number and check that no prime less than it divides the number. You may find it useful to use a product of primes that (just) fit inside a machine word, to reduce the number of divisions needed; in that case you need to take the GCD rather than just dividing.
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Consider the number $143$. This number is composite, not prime, because. $$11 \times 13 = 143.$$ But $$\sqrt[3]{143} \approx 5.229 < 11 < 13,$$ so if you test only possible factors up to $\sqrt[3]{143}$ then you will not check whether $11$ or $13$ divide $143$, and you will not discover that $143$ is not prime.
Usually the best order is trial dividing by the first hundred or so primes, checking if the number is itself prime, then finishing the trial division. For large numbers, do a reasonable amount of trial division, check if the number is prime, run some ECM curves, then factor it completely (with SIQS or NFS, or perhaps even MPQS if the numbers ...
If a number ends in 0, 2, 4, 5, 6 or 8 then it's not prime (except for 2 and 5) If the sum of the digits is a multiple of 3, then the number is not prime (except for 3) Those two rules knock about nearly 75% of numbers. For numbers below 100, the only false positives are $49=7^2$, $77=7\cdot 11$ and $91=7\cdot 13$ which you can learn ...
Even if all the (2-3 billion) computers in the world were working together to prove a number was prime it would still take a 50 digit (or so) number of years to prove a single 512 bit prime. None the less the security industry appears to have a copious supply of large primes numbers at their disposal for cryptographic applications, and the ...
Why would you not just check any given number to see if it is a prime number by simply attempting to divide it by preceding prime numbers? No matter how big the number is there can not be that many you would need to check it against. Other than that unless you know a formula that can tell you what any given prime number may be then there can be ...
Well, by the time you get ready to check a composite number you would have already checked all the prime numbers less than it. And those prime numbers failed. But those numbers include the prime factors of the composite number. If the prime factors failed then the number is going to fail and there is no point in checking it.