How to Check If A Number is Prime?

There are several simple ways to understand whether a number is prime…

                                                          Primility Tests

I]  If P is prime then none of the primes smaller than √P can divide P.

  

II]  If P is prime then P divides  (P-1)!+1 

    Example : for 5  ,(5-1)!+1=25  and 25/5=5   ,  6  , (6-1)!+1=121 and 121≡1 (mod6) so 6 is composite.

III] If P is prime , P divides  A^(P-1) -1 . Here A is a number which can’t be divided with P.

    Ex. 2 ⁽⁴⁾ -1 =15  and  15 /5 = 3 ,  6 ⁴ -1 =1295

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For a little more big numbers, to check primility you can use Dario Alpern’s Alghorithm, below is the link. Write the number and click “factor”, if it says  “is prime”, it is prime. if it is not a prime number, it will write down the factorization of the number. But this alghoritm does not prove if the number is prime for large numbers say hundreds or thousand digits…

https://www.alpertron.com.ar/ECM.HTM

for large numbers like thousand digits, the agoritm may say that it is a prime but actually itwill be a probable prime unless it is certified by prime proving programs.

So what is a probable prime?  A probable prime is possibly a prime which is not certified, which is not known if it is really a prime.

To learn more about prime proving you can visit prime pages :

https://primes.utm.edu/prove/