I can plug numbers in (for k>=1) and see that it works but does this work for all primes? and do all primes have this form? I would like to know where does 6k+1 or 6K+5 come from?
2007-09-11 07:27:07 · 6 answers · asked by sh 1 in Science & Mathematics ➔ Mathematics
Consider numbers of the form 6k + c, if c is 3, then the number is divisible by 3, if c is even the number is divisible by 2. Thus all primes (except 2 and 3) are of the form 6k+1 or 6k+5.
Edit: Thanks, Thomasoa, I fixed it now.
2007-09-11 07:33:45 · answer #1 · answered by Phineas Bogg 6 · 4⤊ 0⤋
It IS true that all primes (greater than 3) have the form 6k+1 or 6k+5.
It is NOT true that all numbers of the form 6k+1 or 6k+5 are prime.
Explanation of the first statement: First notice that EVERY integer (prime or not) falls into one of these categories:
6k
6k+1
6k+2
6k+3
6k+4
6k+5
(Proof: Pick a random number, say 100, then divide it by 6, and look at the remainder (0 through 5). That will tell you which of the six categories your chosen number falls into.)
Now:
The numbers in the "6k" category are not prime (because they're divisible by 6).
The numbers in the "6k+2" category are not prime (except 6(0)+2=2). "6k+2" is the same as "2(3k+1)". That means all "6k+2" numbers are divisible by 2.
The "6k+3" numbers are not prime (except 6(0)+3=3). "6k+3" is the same as "3(2k+1)", so these are all divisibe by 3
The "6k+4" numbers are not prime. "6k+4" is the same as "2(3k+2)"; so all these numbers are divisible by 2.
So, by the process of elimination, the only numbers that can possibly be primes (besides 2 and 3) are the ones in the "6k+1" category or the "6k+5" category. That proves the first statement.
But that does NOT mean that EVERY number that's a "6k+1" or a "6k+5" is necessarily a prime.
For example:
25 is a "6k+1" number (6(4)+1), but it is not prime.
35 is a "6k+5" number (6(5)+5), but it is not prime.
2007-09-11 14:40:42 · answer #2 · answered by RickB 7 · 1⤊ 0⤋
A number must be of one of the following forms:
6k, 6k+1, 6k+2, 6k+3, 6k+4, 6k+5
You don't necessarily have to use 6 to find the forms of a number. You can easily say 3k, 3k+1, 3k+2. But 6 happens to make things easy in this example.
6k is divisible by 6. 6k+2 is divisible by 2. 6k+3 is divisible by 6k+4 is divisible by 4.
6k+1 and 6k+5 does not have a general factor for all values of k. With the exception of 2 and 3, all prime numbers are in this form. However, it does not mean that all numbers of this form are prime. For example,
6*4+1 = 25
6*5+5 = 35
These are not prime.
2007-09-11 14:48:27 · answer #3 · answered by np_rt 4 · 1⤊ 0⤋
Phineas was almost right. All primes other than 2 and 3 are of the form 6k+1 or 6K+5.
That's because all integers are of the form 6K, 6K+1, 6K+2, 6K+3, 6K+4, or 6K+5.
But 6K is divisible by 6, and cannot be prime.
6K+2 and 6K+4 are divisible by 2, so we can only get primes if the value is 2.
6K+3 is divisible by 3, so it can only be prime if it is 3.
2007-09-11 14:42:18 · answer #4 · answered by thomasoa 5 · 3⤊ 0⤋
I did a Google search "6k +1 prime". See if this helps
2007-09-11 14:35:46 · answer #5 · answered by Marvin 4 · 0⤊ 2⤋
no prime nos do not hav a form dats y they r called 'prime'!!!........ur 6k+1 and 6k+5 does not hold on every prime no......
2007-09-11 14:32:16 · answer #6 · answered by GP 2 · 0⤊ 5⤋