Question

Prove that every prime number more than 3 is always in the form 6q-1 or 6q+1, for some
integer 4​

Answer 1

Answer:

Checking a million primes is certainly energetic, but it is not necessary (and just looking at examples can be misleading in mathematics). Here is how to prove your observation: take any integer n greater than 3, and divide it by 6. That is, write

n = 6q + r

where q is a non-negative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5.

If the remainder is 0, 2 or 4, then the number n is divisible by 2, and can not be prime.

If the remainder is 3, then the number n is divisible by 3, and can not be prime.

So if n is prime, then the remainder r is either

1 (and n = 6q + 1 is one more than a multiple of six), or

5 (and n = 6q + 5 = 6(q+1) - 1 is one less than a multiple of six).

Remember that being one more or less than a multiple of six does not make a number prime. We have only shown that all primes other than 2 and 3 (which divide 6) have this form.

Answer 2

5 = 6 *1 - 1        7 = 6 * 1 + 1        11 = 6 * 1 - 1

13 = 6 * 1 + 1        17 = 6 * 3 - 1          19 = 6 * 3 + 1

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 All integers can be written in the form as :

         6 k,  6k + 1 , 6 k + 2 , 6 k + 3, 6 k + 4, 6 k + 5

Each integer from -∞ to ∞  falls in one of the above six forms.  In the above:

    6k is divisible by 6.          6k+2 is divisible by 2.        6 k + 3 is divisible by 3.

     6 k + 4 is divisible by 4.      6k + 5 is same as 6 (k+1) - 1  in the same form as 6 k -1.

Hence, all prime numbers can be expressed in the form:  6k +1 or  6k -1.

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